If Emily was Andy's mom....
Then Jesse would've known because I'm sure Her name would have came up on certain occasions. Leaving Jesse to suspect it was the same Emily who gave her away.
Andy's mom annoyingly barges Into his room to toss things away all the time.so I'm sure she would've noticed the Jessie doll and said something like "I had one if these as a kid".
I'm sure Woody would've know her name since he lives there, and I'm sure he overheard his dad calling her name, regardless if they were married or not, since Woody was a present from him. So it's not like he wasn't around when they were together.