By what mechanism do acids deprotect primary silyl ethers?
Fluoride-mediated deprotection of silyl ethers proceeds through a pentavalent silicon pathway:[1]



This mechanism proceeds the same for both primary and secondary silyl ethers, and there's no reason the mechanism for acid-catalyzed deprotection would be any different. Though I couldn't find a mechanism proposed in literature, it seems likely it would proceed through a similar pentavalent intermediate (Nu = nucleophile):



Steric effects are what drive the chemoselectivity of these reactions. Usually the differences in steric environments around the carbinol carbons is enough, but the use of different silyl groups on the two alcohols can further increase yield of the desired product. These steric effects are particularly pronounced in the following transformation, in which a primary TBS ether is selectively deprotected in the presence of a primary TBDPS ether and a secondary TIPS ether:[2]



Reaction conditions can also have a major effect on the chemoselectivity of desilylation reactions. For example, when HF•pyr is used in a THF-pyridine mixture, selective removal of the primary TBS ether in a protected triol was achieved in 84%. After oxidation of the newly deprotected alcohol to a carboxylic acid, the secondary TBS ether was cleaved using HF•pyr without excess pyridine, leading to lactonisation:[3]



Nelson and Crouch have written a review on the selective deprotection of specific silyl ethers in a compound with multiple silyl protecting groups.[4]



References


Kim, S.-K. Chitin and Chitosan Derivatives: Advances in Drug Discovery and Developments; CRC Press, 2013; p 80. 
Larivée, A.; Unger, J. B.; Thomas, M.; Wirtz, C.; Dubost, C.; Handa, S.; Fürstner, A. The Leiodolide B Puzzle. Angew. Chem. Int. Ed. 2011, 50 (1), 304–309 DOI: 10.1002/anie.201005850.
Körner, M.; Hiersemann, M. Enantioselective Synthesis of the C8−C20 Segment of Curvicollide C. Org. Lett. 2007, 9 (24), 4979–4982 DOI: 10.1021/ol702092h.
Nelson, T. D.; Crouch, R. D. Selective Deprotection of Silyl Ethers. Synthesis 1996, 1996 (09), 1031–1069 DOI: 10.1055/s-1996-4350.