I do screen clients. And I ask more in monetary compensation for my time than most trans guys. I am well aware it's not the norm and I am sure I loose a lot of one-off business to it, however, I'd rather trusted guys that repeat than just have a constant stream of one-offs anyway. I also think I get away with screening being only a momentary hiccup (rather than a deal breaker) because I offer services that a lot of my clients wouldn't feel as comfortable enjoying if I didn't screen. I do find myself pretty often gently reminding guys that not keeping their info private is bad for businesses. If this still doesn't make them feel comfortable enough I tell them about my other job, which is one with a relatively high trust index. Most guys realize that they'd willingly give way more personal information to me without a second thought (like SS#, family contacts, life story, etc) if asked in that context and they realize it's one more piece of info I have to guard and are okay with whatever I need from them after that.
I've gotten clients from friends who refer their friends to me (my friends are pretty awesome & supportive), in which case I can usually go without verifying identity because it's already known and more. Otherwise, I ask for ID (hard to fake or easy to verify stuff: state, passport, military, current large employer ID, currently enrolled student ID) and a health screen from no more than 10 days before we plan to meet where the name on it has to match. The second part is optional, of course, but most guys do it because they realize they're spending like they've had it and plan to use it whether they opt take advantage of that or not.
I have worked as both a TV and trans male escort. I've had a couple scary experiences with both but probably the worst was in LA when I advertised as trans guy and got abducted on an outcall (that started with a meeting in public no less). The dude seemed fine at first, slightly preoccupied, but it seemed to go with the overworked professional white guy white collar Daddy stereotype he obviously had going. I didn't think I needed to screen too hard before, I felt relaxed at dinner, and next thing I know I'm being held at knifepoint in this psycho dude's car (who I also realize nobody I know knows for sure his real info) and he is deciding out loud whether he has a future as a pimp or just wants to take me right to Angeles National Forest. That was years ago and I walked away without a scratch but with the lasting opinion that while most guys don't mean any harm all you need is one malicious dude to ruin your life. I try to encourage everyone regardless of their gender or client base to screen if that's what they need to feel safe. Not screening based on what other people do and what clients want/expect means more people who want to screen won't; it's a self perpetuating fear.