As far as I read it, programs can be loaded on a node and/or on a persona that's loaded on a node. For example, you might have Analyse 4 on your persona, but that won't help your node check on intruders independently unless you load a copy into the node as well. Is that right?

Suppose you hack a guy's 'link. Can you access the programs in his persona as well as those on the node, or are they kept separate from the node at large?

Programs are on Nodes... or in an Agent/IC package.
Personae use the programs on their own node to do things to other nodes.
You cannot use the programs on a target's node, even if you are in that node; UNLESS you are using Server Side Programs, which are different. See Unwired, Server Side Programs Sidebar, Page 109.