Are page/post IDs unique across a whole network?

If I have a page with ID 123, I know my template file page-123.php will be called to render it, but if my theme is used across a network, could a page in a different site also have ID 123?

I assume, if so, that I’ll need to sniff $current_site and requiring an alternate template file if it’s not the site I’m targeting.

I already assume slugs aren’t unique across a network.

Nope, they aren’t unique.

So yes – you’d have to check current site and do that…. or use a theme with a custom menu?