What are the physiological roles of Hydrogen sulfide?
$H_2S$ is the end product of sulfur related respirations (like sulfate respiration, sulfur respiration, etc...).

By aerob (oxygen) respiration the oxygen in $O_2$ has 0 oxidation number, by $CO_2$ the oxygen has -2 oxidation number, so it was reduced while the carbon was oxidized.

By the thiosulfate respiration of Salmonella enterica the following reaction happens by the reduction of thiosulfate: $S_2O_3^{2-} +2H^+ + 2e^- \to HS^- + HSO_3^{-}$. In this case the sulfur in $S_2O_3^{2-}$ has +2 oxidation number while the sulfur in $HS^-$ has -2 oxidation number, so the sulfur was reduced, it was an electron acceptor, just like oxygen by aerob respiration.

So in the case of Salmonella enterica $H_2S$ production is a byproduct of anaerob respiration. It makes growth faster.


  Sulfate-reducing bacteria are those bacteria that can obtain energy by
  oxidizing organic compounds or molecular hydrogen (H2) while reducing
  sulfate (SO2- 4) to hydrogen sulfide (H2S).1 In a sense, these
  organisms "breathe" sulfate rather than oxygen in a form of anaerobic
  respiration.



wikipedia - Sulfate-reducing bacteria



  Salmonella typhimurium produces H2S from thiosulfate or sulfite.



1987 - The phs gene and hydrogen sulfide production by Salmonella typhimurium.
2011 - Thiosulfate Reduction in Salmonella enterica Is Driven by the Proton Motive Force 


S. enterica uses gut inflammation to enhance its sulfur related respiration to outgrow the resident microbes in the intestinal lumen (microbiota). The inflammation creates tetrathionate $S_4O_6^{2-}$ in which the sulfur has an average oxidation number of +2.5. This tetrathionate is reduced by the tetrathionate reductase into thiosulfate with sulfur having +2 oxidation number. So sulfur related respiration helps to make growth faster in order to colonize the gut.


  Here we show that reactive oxygen species generated during
  inflammation react with endogenous, luminal sulphur compounds
  (thiosulphate) to form a new respiratory electron acceptor,
  tetrathionate. The genes conferring the ability to use tetrathionate
  as an electron acceptor produce a growth advantage for S. Typhimurium
  over the competing microbiota in the lumen of the inflamed gut. We
  conclude that S. Typhimurium virulence factors induce host-driven
  production of a new electron acceptor that allows the pathogen to use
  respiration to compete with fermenting gut microbes. Thus the ability
  to trigger intestinal inflammation is crucial for the biology of this
  diarrhoeal pathogen.



2010 - Gut inflammation provides a respiratory electron acceptor for Salmonella
2001 - The Alternative Electron Acceptor Tetrathionate Supports B12-Dependent Anaerobic Growth of Salmonella enterica Serovar Typhimurium on Ethanolamine or 1,2-Propanediol 


Since $H_2S$ is a gasotransmitter in the human body, there can be other mechanisms which help S. enterica.


in small amounts $H_2S$ has anti-inflammatory and anti-apoptotic effects
in large amounts $H_2S$ has pro-inflammatory and pro-apoptotic effects


So S. enterica can probably cause inflammation due to killing cells with a fast release of $H_2S$ or prevent inflammation and keep infected cells alive with a slow release of $H_2S$. I found many evidence of the pro-inflammatory theory. By the anti-apoptotic theory I wasn't so lucky, I found only a single review about anti-apoptotic strategies of intracellular pathogens, but it did not mention $H_2S$ production as a possible mechanism. So it might not be true, further studies needed...


  In the digestive system, H2S exerts potent anti-inflammatory actions,
  regulates blood flow and smooth muscle tone, modulates epithelial
  secretion and promotes healing of ulcers [4, 5].



2012 - Hydrogen Sulfide: A Rescue Molecule for Mucosal Defence and Repair



  Hydrogen sulfide (H2S) is the most recent endogenous gasotransmitter
  that has been reported to serve many physiological and pathological
  functions in different tissues. Studies over the past decade have
  revealed that H2S can be synthesized through numerous pathways and its
  bioavailability regulated through its conversion into different
  biochemical forms. H2S exerts its biological effects in various
  manners including redox regulation of protein and small molecular
  weight thiols, polysulfides, thiosulfate/sulfite, iron-sulfur cluster
  proteins, and anti-oxidant properties that affect multiple cellular
  and molecular responses.
  
  Understanding precise pathophysiological signaling mechanisms and the
  metabolism of H2S is a topic of active research. Unraveling H2S
  interactions within different tissues, with other biochemical
  molecules and various signaling mediators is becoming ever more
  complex.
  
  These results demonstrate that H2S donors can down-regulate adhesion
  molecule and proinflammatory cytokine expression, therefore
  identifying H2S, its synthesis enzymes, and molecular targets (e.g.,
  KATP channels) as potential targets for novel anti-inflammatory
  therapies.
  
  Thus, all of the above findings demonstrate that H2S induces
  cytoprotection by an anti-apoptotic pathway.



2013 - Hydrogen sulfide chemical biology: Pathophysiological roles and detection



  A short course of H2S infusion was associated with reduction of lung
  and kidney injury. Prolonged infusion did not enhance protection.
  Systemically, infusion of H2S increased both the pro-inflammatory
  response during endotoxemia, as demonstrated by increased TNF-α
  levels, as well as the anti-inflammatory response, as demonstrated by
  increased IL-10 levels. In LPS-stimulated whole blood of healthy
  volunteers, co-incubation with H2S had solely anti-inflammatory
  effects, resulting in decreased TNF-α levels and increased IL-10
  levels. Co-incubation with a neutralizing IL-10 antibody partly
  abrogated the decrease in TNF-α levels. In conclusion, a short course
  of H2S infusion reduced organ injury during endotoxemia, at least in
  part via upregulation of IL-10.



2012 - A short course of infusion of a hydrogen sulfide-donor attenuates endotoxemia induced organ injury via stimulation of anti-inflammatory pathways, with no additional protection from prolonged infusion



  H2S causes apoptosis in HPSCs by activating the mitochondrial pathway.
  It is suggested that H2S might be one of the factors modifying the
  pathogenesis of pulpitis by causing loss of viability of HPSCs through
  apoptosis.



2011 - Hydrogen Sulfide Causes Apoptosis in Human Pulp Stem Cells



  The level ofendogenous H2S was increasing along with the infection
  occurrence and the gradient of infection aggravate. We can presume
  that endogenous H2S participated in inflammatory reaction of abdominal
  infection and could be one of the serology index which concerned with
  the gradient of infection.



2012 - P53 The level changes and clinical significance of endogenous hydrogen sulfide of patients with acute abdominal infection



  The evidences showed that H2S has an obvious effect on colon smooth
  muscle contraction, and can increase the intestinal movements in slow
  transmit constipation. Our experiment states that H2S has
  anti-inflammation effect in prophase of acute peritoneal cavity
  infection.



2014 - P48 Regulation of hydrogen sulfide in digestive stystem
2010 - Hydrogen Sulfide Improves Neutrophil Migration and Survival in Sepsis via K+ATP Channel Activation
2012 - Hydrogen sulfide and resolution of acute inflammation: A comparative study utilizing a novel fluorescent probe
2008 - Staying alive: bacterial inhibition of apoptosis during infection



  H2S is believed to have two contradicting roles in inflammation. It
  acts as both pro- and anti-inflammatory molecule(9). Li et al.
  reported that the physiological concentration of H2S has
  anti-inflammatory effects, while higher concentrations of H2S can
  produce pro-inflammatory effects(10). The H2S inflammatory role was
  also studied in different systems. In the gastrointestinal tract, the
  H2S regulating role functions by activating KATP channels in order to
  promote the inflammation response(57). The similar H2S function was
  observed in pancreas(7), but the actual mechanisms are largely
  unknown. In conclusion, H2S pathway is a possible route for targeting
  the inflammation treatment. However, much work needs to be done for
  understanding the mechanisms of the contradictory roles of H2S in
  inflammation.



2012 - The Crosstalk between H2S and NO Signaling Pathways
2013 - Gasotransmitters, poisons, and antimicrobials: it’s a gas, gas, gas!



  Developing evidence suggests that dysbiosis (abnormal microbial
  composition or function) can contribute to if not cause chronic
  intestinal inflammation. 5,7 This inflammation can be caused either by
  an abnormal composition of entericbacteria with an elevated ratio of
  aggressive vs protective species, defective production of short-chain
  fatty acids and other protective microbial products, or enhanced
  production of hydrogen sulfide and nitrates that block butyrate
  metabolism and disrupt the mucosal barrier.



2010 - Inflammation and Nutrition in Chronic Disease
2013 - Potential Role of Hydrogen Sulfide in the Pathogenesis of Vascular Dysfunction in Septic Shock



  These results showed that physiological concentrations of H2S can
  induce apoptosis of PDL cells and HGFs in periodontitis, suggesting
  that H2S may play an important role in periodontal tissue damage in
  periodontal diseases.



2009 - Hydrogen sulfide induces apoptosis in human periodontium cells
2010 - Bacteria-derived hydrogen sulfide promotes IL-8 production from epithelial cells



  We have shown that inactivation of H2S producing enzymes
  (cystathionine beta-synthase, cystathionine gamma lyase, or
  3-mercaptopyruvate sulfurtransferase) and NO-synthase in several Gram
  (+) and Gram (−) bacteria render them highly sensitive to different
  classes of antibiotics (Gusarov et al., Science 325 (2009) 1380–1384;
  Shatalin et al. Science 334 (2011) 986–990). We also presented
  evidence that Bacillus anthracis-derived NO is critical at the early
  stage of infection (Shatalin et al. PNAS 105 (2008) 1009–1013). Here
  we show that: (1) cbs/cse and nos mutations change Bacilli global gene
  transcription profile; (2) apore formation process in cbs/cse and nos
  mutants of B. anthracis is affected; (3) virulence of cbs/cse and nos
  mutants of B. anthracis is diminished. These results demonstrate that
  bacterial H2S and NO are an important virulence factors, and that
  enzymes generated these gases may serve as an attractive target for
  antimicrobial therapy.



2014 - S7-6 Role of H2S and NO in Bacillus anthracis spore formation and virulence


Btw. there is non-hydrogen sulfide producing S. enterica too, which can probably (no study about this yet) cause salmonellosis. So using thiosulfate as electron acceptor and producing $H_2S$ might not be essential by the infection. (There are other non-sulfur electron acceptors e.g. nitrate, fumarate, etc... for the case of anaerob metabolism.) 


2013 - Genetic Analysis of Non-Hydrogen Sulfide-Producing Salmonella enterica Serovar Typhimurium and S. enterica Serovar Infantis Isolates in Japan
2011 - Salmonella enterica Serovar Typhimurium Colonizing the Lumen of the Chicken Intestine Grows Slowly and Upregulates a Unique Set of Virulence and Metabolism Genes
2013 - Salmonella Uses Energy Taxis to Benefit from Intestinal Inflammation
2006 - Role of Gluconeogenesis and the Tricarboxylic Acid Cycle in the Virulence of Salmonella enterica Serovar Typhimurium in BALB/c Mice 
1981 - Aerotaxis in Salmonella typhimurium: role of electron transport.
2010 - IDENTIFICATION OF NOVEL VIRULENCE GENES OF SALMONELLA ENTERICA
2014 - Increasing prevalence of hydrogen sulfide negative Salmonella in retail meats


Overall hydrogen-sulfide and other gasotransmitters are important virulence factors of many pathogens.