I think one can get a lot better at screens. They are tough in pickup b/c you aren't running an offensive set (I've never run an offensive set but I imagine picks would work way better b/c they are actually part of the set) so the guy with the ball may not ever use picks, may not be ready to use it, may not execute it well.
If you set a pick, for sure try to set it perpendicular to the defensive guy's shoulder - this makes it hard to get around, I see a lot of lazy picks that are kind of behind a guy and it's easy to slide over such a pick. Although sometimes I set a front pick to set up a long-range shot (I love doing this in transition) and of course that involves just stopping in front of someone, not perpendicular to them at all.
You should also be constantly screening away from the ball, especially on the baseline so your teammates can flash to the other side of the rim down low. Also pinning a defender so a guy can break free to go shoot a 3 will work well too.
On defense you need to anticipate these types of screens and get more used to either running through them (if the offense isn't set, F them imo, although I almost hurt a guy running him over - altho I didn't see the screen at all) or avoiding them: deciding quickly whether to go over or under...or sometimes play off your man so you can easily run around one. One guy who plays in the pickup game I'm in is huge (just huge) so all he does really is set moving screens and it's sometimes impossible for me to get around (he is really really large).
As for moving screens, yeah, I don't call them (no one really does) but if it's bad and someone gets shoved or knocked over it will get called. I also will warn guys if they keep setting moving screens (same with 3 seconds on offense) and just say something like "that is another moving screen, I'll start calling that" although I never start calling it.
Tbh, in pickup you should just set moving screens, there's no reason not to really b/c they don't get called - hell, they rarely get called in the NBA. Also I guess there is a difference between a hugely moving screen and a little shuffler here or there where the guy setting the screen is technically not completely set but getting set (not completely moving when the screen is set).
If a guy is moving and screening you, initiate contact and then I think you can call it if it's basically boxing you out of the play. Sometimes running down court I'll kind of get in front of someone to mess them up but if I just continued to do it as they tried to run through me, it's a foul on me and I'll likely get hurt (as they'll be running through me). I think once some guy did this to me essentially and I tried to run around *hard* - he was pissed but I told him he was setting a clear moving screen and not to do that if he didn't want to be knocked down.
Oh, one last thing - when setting good screens I try to lean into them a bit (not with my elbow or shoulder or something dirty) simply b/c a good screen will catch someone off guard and it's very easy to get knocked over or knocked back and then you're off balance. So remember to bend your knees a bit and get your weight shifted forwards!