The effects of online chat on written language skills With the advent of texting and online social networking. There has been a dictionary of new words and acronyms, based on abbreviations. I have noticed that I will misuse there, they're and their, hear and here... and the incidence of my grammatical errors has increased with the increase in longevity of my online and text communication. 

I understand this is anecdotal, but I am wondering:
Are there studies showing the effects of online social networking and texting on people's  written language skills?
The research on this topic is limited; it seems, however,  that the advent of online and mobile communication may assist in improving literacy skills of children and people learning a second language.  

The use of phonetic abbreviations, assists the learner in developing the connection between verbal and written language. The use of acronyms relies on the users knowledge of how words are spelt. Which requires a presumption of literacy.



Although some studies are inconclusive, the trend shows that texting and online chat assists in child literacy. Enforced limitations on text size, and the use of a, rapidly, evolving language, promotes creativity in the use of written language.


  Recognising that many, if not most, textisms are some form of phonetic
  abbreviation, Plester, Wood and Joshi (2009) argue that to produce and
  read such abbreviations requires, in the texter, a level of
  phonological awareness (and orthographic awareness). While spelled
  incorrectly in a conventional sense, many textisms are phonologically
  acceptable forms of written English. Decades of research has
  demonstrated a consistent association between different forms of
  3phonological awareness and reading attainment.  David Crystal,
  honorary professor of linguistics at the University of Wales, believes
  that sending frequent texts can actually help children to read and
  write because of the abbreviations used. “People have always used
  abbreviations ... They do not actually use that many in texts but when
  they do they are using them in new, playful and imaginative ways that
  benefit literacy” (Leake, 2008).
  
  Crystal (2008) believes that the widespread concern about the impact
  of texting on children’s literacy is unfounded. The brevity of the
  text style, and the 160 character constraint of an SMS, requires the
  author to write economically, inventively and playfully – doing this
  is good practice when learning to read and write. Wood, Plester and
  Bowyer (2008) concur that “when texting, the children have the freedom
  to ‘play’ with the construction of language that they are learning
  about at school, and are creative in their use of it. They also have
  regular engagement with it.” Plester, Wood and Joshi (2009) believe
  that any engagement with the written word (as opposed to the spoken
  word) – including reading and writing textisms in digital form on
  mobile phones – is beneficial for children. Wood, Plester and Bowyer
  (2008) posit that “children’s use of this technology appears to have a
  positive impact on their developing literacy, as it provides children
  with an additional resource for learning about and experimenting with
  letter-sound correspondences and language, and for reading and
  ‘decoding’ text.” They conclude that “If our children are showing
  difficulties with reading and spelling attainment, it would seem that
  this is in spite of the contribution of textism use, not because of
  it.”    


....


  Evolution of language Language changes constantly. To illustrate this
  point, at the beginning of each school year Cindi Rigsbee, a sixth-
  and seventh-grade reading-resource teacher in the USA, shows her
  learners texts from Old English, Middle English, contemporary English
  from the time of Jane Eyre, and a MySpace page. Throughout the year,
  Rigsbee often refers back to this lesson to remind her learners of the
  different forms of writing (Bernard, 2008). Other teachers have
  contrasted IM lingo with Shakespeare to demonstrate how English has
  evolved (Lee, 2002)


(The effects of texting on literacy: Modern scourge or opportunity? - Shuttleworth Foundation)



The use of texting and online chat has also promoted Second Language Acquisition (SLA). The emphasis being on the written language, and the inability to utilize body language, and facial gestures, as a bridge to overcoming learning difficulties.


  It was found that text-based online chat promotes noticing more than
  face-to-face conversations, especially in terms of learners’ noticing
  of their own linguistic mistakes.


....


  Philp’s (1999) findings pushed the agenda of research on noticing away
  from general evidentiary studies on the positive effect of noticing on
  SLA  towards exploratory studies on contextual factors that affect
  noticing. Realizing the potential effect of learners’ memory capacity
  on noticing as laid out in Doughty’s (2001) cognitive comparison
  framework, Mackey, Philp, Egi, Fujii, and Tatsumi (2002) designed a
  study to explore the relationship between learners’ working memory
  (WM) capacities and their noticing of interactional feedback. They
  found that there was a positive relationship between WM capacities and
  their noticing of interactional feedback, and that learners at lower
  developmental levels exhibited more   noticing than those at higher
  levels.


(NOTICING AND TEXT-BASED CHAT, Chun Lai and Yong Zhao, Michigan State University)



It is unclear what the advent of technological communication has upon the literacy of   adult native language. Any links to current research on this would be welcome.

Note I would like to acknowledge that I have used the links provided in the comments to answer this question.

Will update answer upon acquisition of new material.