How to share Outlook or Owa contacts to another user on my organization

There isn't a way to share them user to user that I know of, however you can export and import them to CSV. The benefit of this is your users can manage it on their own, however you (or they) won't have any central administration over the contacts.

If you're feeling particularly old school, you could even create a Public Folder and assign permissions to the required users, though Public Folders seem to be on the way out in favour of Office 365 Groups.

Hope it helps,
Daniel