Studdering Issue with Other Child at the Babysitter's House

While I know your son does not have a speech problem I thought I would chip in with advice on helping children who do have speech problems... this may work for your son (only a lot faster!) In my child development degree we learned that if a child says my two feets or my two foots instead of saying no we say feet... simply repeat it back the correct way "your feet" or "oh do you mean feet?" Correcing by example is much more effective then correcting by pointing out what the child did "wrong". So for your son when he stutters simply repeat the sentance back to him, almost to clarify.
I would also like to add that I definitely think you should talk to the childcare provider. She may not be aware that it is carrying over to your son, or she may be at a loss of what to do. I have an in home childcare and I can honestly say that we need the parents to communicate with us and keep us posted just as much as we have to keep them posted as well!
If you would like any further advice on this please feel free to contact me. Give it time... I'm sure it will get better!

When your son studders I would simply explain that speaking that way is something that little boy has to work on just like your son has to work on...(fill in something that your son is working on and getting better at or something he had to work on to overcome such as colors, numbers, letters, etc.)Explain that his repeating that is actually not very nice because it may make the other boy feel bad. Good Luck

Instead of saying dont talk like so and so...tell him to slow down and say that again please. When my step-son developed the stuttering we would do that and it aleviated the problem.

I have insight on this on 2 levels. One my step-son has a slight speech impediment that has gotten better over the years, and it rubbed off on the nieghbor kid, and that went away after we told the boy that he might be hurting some feelings. But the other one was with my youngest, she started to studder around that age. We came to find out that she was just so excited about learning new things and ways to express herself, that she could get it out fast enough. So it might be just that simple. With that one, all we did was tell her to slow down, that we had all the time in the world to listen to her, and that one went away too. Good luck.

A work of caution: If you make an issue out of it your son will do it more often to get a reaction out of you. When he does imitate his friend ask him to slow down because you can't understand him when he doesn't use his own voice. Also, paying too much attention to this phase of imitation may actually cause a problem. He should also not be making fun of the other child or correcting him.
It sounds like you should make the trip to that town to meet that child and even their parents. Have a play date, avoid the stuttering topic and observe quietly. Just use good judgement.