A good personal trainer will know the difference between the pain of exertion (eg muscles burning when you're doing squats and lunges or lifting weights) and the pain of injury (eg joint pain or shin splints). My personal trainer will tell me to 'push through it' if my muscles are burning, but if I say 'That's really hurting my knees' or 'I can really feel that in my lower back' he'd switch to a different exercise or make adjustments to take the strain off joints etc. I probably say 'F*ing hell, that hurts!' at least twice a session and my trainer knows I can still work harder, but if I say 'Hang on, I can feel my Achilles pulling...' he'll get me to stop.
Similarly, it's really normal (and positive) for your muscles to be aching like hell for a day or two after a really tough session. That's how you build new muscle and tone up. But it's not normal to have a swollen knee or ankle or any sharp pain that feels like a localised injury.
So, I think it probably depends how you express the pain you're in. 'Ow, it hurts, I can't do it,' is something I think most trainers would expect you to push through - it is their job to push the client, as that's how fitness is improved. But if you said 'Hang on, I think I've injured my knee' then that's a different matter. Your trainer should be attentive to any potential injuries, but equally he won't be able to read your mind.
Of course, if you're not comfortable with him or his training style, absolutely do find someone else, though, regardless of the injury/pain issue. You do need someone you feel comfortable with and can trust, and sometimes it's just a matter of finding someone with the right personality. My PT is really positive and gives his clients loads of praise, which works well for me, whereas my friend prefers to go to a terrifying woman who literally shouts at her like an aggressive drill sergeant.