My DD1 was exactly the same.
A friends DD is 18 months younger & when they were little she was a rat & used to always be into my DD.
I never worried about it though. To me it says that DD1 knows right from wrong. Pushing, hitting other kids & snatching toys is wrong & something these other kids still need to learn.
I was quite proud of my DD. Very p*ssed off with the other childs parents, but I would sometimes say 'no ---------. Jaz was playing with that' & give the other child a different toy & try to distract them instead.
I think you'll find if you do that the other mother will tend to agree with you & not say anything. If she does I'd be leaving!