Faith - nothing more, nothing less - in the belief that the person really does have a special connection to divinity.
While there may be prescriptions within various religions as to what actually constitutes a prophet or not, there have certainly been a number of people after who set up significantly sized followings based on their own claims.
Heck, look at Christianity - Judaism does not accept that Jesus passes all the tests to be regarded as a major prophet, but it certainly hasn't stop that religion.
The Prophetic experience is a very powerful one, though. Been through it myself, and I could probably still declare myself a prophet and build up a following if I really wanted to, simply on the basis of my past experiences, connections, insights.
But - one of my first spiritual lessons was that of humility, that we are all equal - that to stand apart and above others is very unspiritual - what does bringing attention on the self feed other than the ego? What right does any to claim titles unearned?
There are more subtle ways of playing the system, to get your own message out there - and allow it to be judged on whatever merits or no it may have.
If anyone declares themselves a prophet, it's probably because the spiritual experiences they are going through are so powerful - if your consciousness feels incredibly expanded and made privvy to uncommon knowledge and insights, all within a powerful feelings of spiritual experience, then you rationalise that who else could be capable of receiving these, if not a Prophet?
Yet the title of Prophet I think is one best not declared, but earned, and that peers, humanity, history, become the real judge of who is or may be decided to be a prophet, not based on claims, but by action.
And that, I think is the necessary fruit for anyone claiming to be connected to Divinity, to any degree, at any level, whether claiming prophet hood, or even simple belief. Without works, we give nothing.
2c.