Who exactly has to authorize an ex president to talk to foreign leaders?

Only the sitting US President may set US foreign policy, which is generally through the State Department and with some oversight from Congress. Not an ex-President. Not a candidate for President.

Everyone is more than welcome to express their opinion on foreign affairs. Any candidate for President should be ready, willing, and able to discuss what they would do if elected. But no one outside of the current administration may actually conduct foreign affairs.

So the real question boils down to what Trump talked about with these foreign leaders. If he were negotiating with these foreign leaders about the US policies he would implement and asking for what they would offer in return, that's an illegal conduct of foreign affairs. On the other hand, if he were hosting something like a birthday party that had lots of attendees and no private meetings with the leader, that would be fine. The problem is that these were private meetings AND we don't know what he talked about. I'm not aware of any transcripts or summaries of the discussions at these meetings. And based on his history, it is not at all unreasonable to ask if he were conducting foreign affairs. Trump has no filters that would prevent him from breaking the law as long as he felt it would be in his best interest to do so. His own account of the meetings is simply unreliable, as he lies almost as often as he breathes.

Did he engage in an illegal conduct of foreign affairs? We just don't know clearly one way or the other. It's not clear that he did, and it's not clear that he didn't. Hence the suspicion and questions about the meetings.

--Peter