My child was around 2.5 when we had him evaluated. We actually had one done through early intervention and then 2 done on our own. The early intervention one covered more than just speech. I was in the room with him answering questions while another person was testing my son. The tests were play based.
The speech only evaluations I had done were also play based. Some of the tests were geared toward testing the child's understanding so it's not required that the child speak a lot. Sometimes the testing involved repetition to see if that would draw the child into speaking or, in the case of a younger child, demonstrating the behavior. One therapist pulled out some bubbles to try to watch my son's mouth movement in order to test coordination and muscle movement of the mouth.
I was worried my son wouldn't talk at all either, but he did well and they were able to hear some of the things that I was concerned about. For the most part, I was in the room with him, but managed to leave the room for part of one evaluation because my son was interacting too much with me and not her. But if your child doesn't want you to leave there should be no reason that you shouldn't be able to stay. My suggestion is to try not to engage him while the therapist is doing the evaluation unless the therapist requests it. If my son came over to me, I wouldn't engage in play but instead would redirect him back to the therapist.
My son ended up with a borderline need for speech therapy. We did it for about 6 months and it made a huge difference.
For the regular sessions, I initially went into the room with him but stayed off to the side. I did my best to redirect him back to the therapist when he came over to me and I also avoided trying to interpret what he was saying because I wanted the therapist to not be affected by what I knew of his speaking ability. Eventually, I worked up to staying in the waiting room for part of the session and then he'd come get me. Later I could stay in the waiting room the whole time. I also "escaped" a few times in the beginning by saying I had to go to the bathroom and I'd be back in a bit. Then I'd leave and come back in 5-10 minutes and pop my head in. If he didn't miss me, I'd stay in the waiting room.