I generally either disable email or configure Discourse such that it can’t send email. They will get summary emails if you’re not careful.
You probably want to turn on login required and/or the don’t-index-this-site (I forget the name) system variables so that Google doesn’t index it too.
They’ll need to do a “forgot password” (but you might check out Migrated password hashes support) to get connected with their old account. If someone tries to register for an account with an email address that already exists they won’t be able to since the email address is already a user.
Some number of people will have forgotten what email address they used when they created their account and will whine about it. You’ll have to get them to convince you that they are that person and change the email address for them.
We generally recommend users to use their email address to log in to the new forum for the first time because the import changes a (generally very small percentage) of usernames because the Discourse username rules are more strict.
So if someone has forgotten the email address they’ve registered with, they will be able to login with their username/password in 99% of the cases if you are using the migrate paasword plugin. And in the rare case they forgot their email address and their username belongs to one of the few that were changed, you’ll only need to tell them the new username that corresponds with the old username. No convincing needed.