I'm a DIYer and I purchase a fair amount of dive gear online or "used." The first thing I do when I get new gear is make sure that it's assembled properly and read the manufacturer's instruction manual.
Pool access isn't required to test out new dive gear. I've never tested out new or new-to-me gear in a pool. I've always done comprehensive dry land testing and then taken the gear out for real world testing in benign conditions in the ocean. Relatively shallow shore dives are ideal for testing out new regs/BCDs/drysuits.
When I first take possession of a piece of gear, I test it out on dry land first. That means that
every single feature is checked out. These checks are identical to those that I perform during my pre-dive routine. Such checks should have been taught to you in your basic OW class.
For a BCD, look it over visually then test it functionally. Are there any rips/tears in the material? Are any plastic buckles broken? Are all the D-rings intact? Do all of the zippers work? Does the velcro hold securely? Is it adjusted properly for me? Does the inner bladder hold air for an extended period of time? Does the power inflater assembly inflate/deflate appropriately (inflate
only when ON button is pressed; deflate
only when exhaust button is pressed)? Do all of the dump valves activate/deactivate properly? Does the overpressure valve activate properly when the BCD is over-inflated? Do the tank cambands hold a tank securely?
For regs, you should be doing a set of comprehensive, functional tests -- the same set that you should be doing before
every dive.
couv et al. have put together a nice checklist here on ScubaBoard.
At a minimum, you really need to do a cursory visual check, listen for leaks, do a vacuum test, and make sure that i.p. (must measure i.p. with an i.p. gauge) and cracking pressure (subjective breathing test) are reasonable. Bubble checks can be done by attaching the first stage to a tank, pressurizing the system, and submerging the first stage in a small bucket of water. You can also stick your face into some water to verify that the second stage breathes dry.
The vast majority of divers either don't know how to conduct these functional gear tests or are too lazy to do them on a regular basis. Unfortunately, most instructors don't teach their students how to do these checks. (Huge oversight, IMHO, and may indicate that a fair number of instructors don't have an in-depth understanding of how to really troubleshoot reg issues.) Since most beginning divers rent their regs, they should be doing these tests whenever they take possession of a rental reg.
These tests should be done before the rental reg even leaves the rental shop. Doing the tests right away gives the rental shop an opportunity to fix any issues on the spot or replace the reg altogether.