> At what point would they still be considered the old company's property?

It depends on your contracts (aka NDAs). Sometimes employers do ban for that very same reason from joining a competitor(s) for x years from date of separation

It also depends on your law. In California, some companies still try to put these restrictions into contracts, but that contradicts state law so it has no legal effect. The intended effect is intimidation, however, which may work with some people.