Technically, you can accuse people with no evidence at all. The question is whether the accusation should be taken seriously.
I find “evidence” to be a better word than “proof” here, because there is so rarely “proof” of anything. There is conflicting evidence, and we decide how to weight the evidence.
For example, we wanted proof that Obama was born in the USA. His campaign put out a “birth certificate” which it claimed was proof. But people analysed the paperwork, and many think it was bogus, or that it didn’t show he was actually born here — it was evidence, but not “proof”.
Anyway, there was evidence that the campaign was paying rent on the townhome. They didn’t make up the accusation our of whole cloth — they had the FEC filings. I don’t know when the burden should shift from the accuser to the accused — but I know that when it’s a conservative, they better well jump to it, because the media isn’t fair.
Being involved in politics as you seem to be, you should know as much as anybody that politics isn’t about proof, or evidence, or facts. It is about perception — the perception of the voters. If you can’t convince 50%+1 voters to pick you, it makes no difference WHY. It makes no difference if you can prove charges against you were false. You need to win over the voters. Nobody is going to win the voters over for you.