Does screen sharing (System Preferences > Sharing > Screen Sharing) work when the computers are on different networks? For example, if computer A is using Wifi in Texas and computer B is using Wifi in New York, will this still work?
Yes. And you can make it work through most firewalls if you're both using Messages. Messages will broker the handshake to start the session and then turn it over to the screen sharing app.
Start up Messages and make sure Screen Sharing is enabled:
Find your buddy in your contact list in Messages and click on them. Then select
Buddies -> Share My Screen with Buddy... or
Buddies -> Ask to Share Buddy's Screen... from the menu bar to initiate a screen share session. You can share control with these sessions.
Yes, as long as you port forward (
VNC default port is 5900, so make sure that your router is correctly port forwarding this. You can change the port in your router settings.
This Apple KB article gives more details about OS X screen sharing.
If you are trying to access one of your own computers that uses your same Apple ID, the best method is to use Back to My Mac, as long as you are on a Mac that has not yet been updated to Mojave. Back to My Mac has now been discontinued by Apple.
You have to make sure you enable "Back to My Mac" in the iCloud Preference Pane on both computers:
Under the Sharing Preferences pane disable Remote Management if it is enabled, and then enable Screen Sharing.
Wait a moment and your computer should show up in Finder under the Shared section.
If you don't see a "Shared" section or the computer doesn't appear there, click on Finder and then open Finder's preferences (⌘+,) and make sure that everything in the Shared section is selected: