DESPP generally enters the information into a database, which is available to LE. If a police officer contacts dispatch to run your license (e.g. a wants and warrants check), dispatch will also be able to pass along (fairly promptly) that the subject possesses a pistol permit. There are also numerous instances of LE checking guns they encounter against those associated with an individual. In the past, a state report indicated that SLFU retains the hard copy forms "indefinitely." As a matter of practicality, most are entered/ retained. I can speak from experience when I say that I have purchased firearms that have had a DPS-3 sent in but which were not entered into the database. IN the past, some people also experienced situations (when DPS tried to reconcile records) when it could reasonably be surmised that forms had been lost/ not retained because many sellers were contacted seeking forms that were purportedly not received- in some instances well over 5 years prior (the statutorily prescribed period for which the seller must retain a DPS-3).
What towns do varies, a lot. Some towns have been suggested to keep parallel records on gun ownership, based on permit applications and DPS-3 forms. Other towns compile information from other sources (potentially too) New Haven, for example, requires (possibly required) permit applicants to provide lists of firearms owned by them.