I will have to provide usernames and passwords to the other domains admin. This doesn't seems right as they will know the passwords of my users. The other way I can think of is to copy my user's profiles and provide them. Is this correct?
I dont know what you are asking about sharing PW, if you got a user called bob under domain A and you want him to access a share in domain B in your DC in B should have an account called bob in order for him to access that share..
You dont need to share passwords just create users that you want to access that domain and when they supply the right creds they can access it