Wire does not require a phone number. It does allow you to provide one if you like.

With Signal, you need the phone number later because the phone number is the only way to add people. Also, note that you may have to enter the phone exactly as they originally entered it in order for it to be found. I know that's how What'sApp works, and with What'sApp, you can't even start the conversation from within the app. You've got to leave the app and add them to your contacts first, then go back to the app and search to see if it found them, and then go back and edit the contact to twiddle the formatting of the phone number if it didn't, until it finally does or you give up. It's pretty much the worst UX imaginable, I have no idea why people put up with it. At least Signal lets you do it all from within the app so it takes like a tenth of the time and effort.