It depends on the field and rules, really.
At one local field, if you call a paint check on yourself, you cannot move or shoot until the ref checks you. Opponents can move on you, though. So, unless you can get a very nearby ref's attention quietly, all that calling a check on yourself does is get yourself bunkered easily. (I personally find this to be a completely silly way of handling things, as it just promotes playing on if you're unsure of whether it broke or not.)
At the one I play at most of the time, you call for a paint check and keep playing/defending yourself while being checked. It's up to you to stay behind cover. Last weekend I shot a kid who was fully exposed broadside to me. I couldn't tell if it broke, he called for a paint check, I shot him again while the ref was moving toward him. By then the ref's there checking him, he's still TOTALLY exposed, and I still can't tell if either of the first two shots broke (it was all the way across the field), so I shot him two more times while the ref was right there next to him, at which point the ref obviously saw one break and pulled him. Was it cheap to shoot him while he was being checked? Maybe. But it's clearly in the rules briefing at the start of the day that you're not off-limits just because you ask for a check. He should have at the very least shifted behind the bunker instead of continuing to sit in the open next to it.
Granted, both of these examples are of calling a check on yourself. At the former field, I've never heard anyone call a check on an opponent, and at the latter, it's still the same thing; call for him to be checked, but he's still fully alive and free to shoot at you while being checked.