This will sound kinda odd, but...
If the puck is outside of the circles, I have had a good amount of sucess just moving in front of / through the screener. Remember, you have forty pounds of equipment on, most skaters can not stop you shouldering through. And I have never been called for shouldering someone out of the way, just don't cross check them or anything like tackling them. I was warned once for putting the face of my blocker in the guy's back to push him out.
This works particularly well in the lower levels that I play at. The screener rarely knows how to react and just stands behind you. If you have to drop back they normally move or they end up in your crease (so the other team wouldn't get the goal anyways)
I would expect this is not a smart move in upper levels, I'm sure you would pay somehow.
Oh, and not that I ever do this...

but if the puck is in close then the guy screening me was simply in the way of my attempt to butterfly slide out to cut off the angles.

That will stop them camping in front of the net (I've heard...)
