As a general rule, if you can catch "it" (for whatever specific value of "it" you care to use) by non-sexual contact with a critter, you can catch it from them via sex. Exceptions exist, including things like the Flu, or similar respiratory infections. Going the other direction, the same holds pretty much true, too.
As far as specific STDs like the clap, syphilis, Herpes, AIDS and the like, while the critter could be contaminated with them (as has been noted, perhaps by someone who has the disease "gettin' jiggy wit' 'em" before you come along) thanks to differing chemistries, body temperatures, and several other factors, it's EXTREMELY unlikely that the critter will actually "have the disease", but it's possible that they're "contaminated" with it, and dipping your wick in 'em (or letting them dip their wick in you) a short enough time after the infected person who "contaminated" them would expose you to the relevant bugs. The risk that you're going to come down with a case of the clap (as a ferinstance) from fucking a dog that some guy who had it has been with is *HIGHLY* dependent on how long before you he was with the critter in question - MOST of the bugs that cause STDs don't survive for more than a few hours in a critter other than a human.
As far as random diseases, the risk is so highly variable that trying to make any prediction is pretty much an exercise in futility.