For truly private jet flights it can be trickier. Many truly private jets operate as private flight operations under part 91 of the FAA’s rules. That has a lot less red tape for a private operator but means they truly need to be a private operator and can’t pickup others for hire.
Several startups have tried to get around the above via legal loopholes via ‘Uber for airplanes’ models and got shot down by the FAA.
However for small planes (Cessna etc.) pilots have to reach their flight hours and for financing that take passengers. Sometimes just for scenic round-trips. Sometimes over longer distances. Examples for sites arranging this (German examples) are https://flyt.club/ https://www.mitflugzentrale.de/ or international https://www.wingly.io/
That's not really Uber for planes, but it gets you places in a way that is often much more expensive otherwise.
If you can't, you'll find that you're really just buying into hype more than actual productive travel. All of the arrangements will have strict conditions on passengers, luggage, arrival/departure times and many trips will only be 1-way so you're left finding your own way back, and that might be from small regional airport somewhere.
[1] https://en.wikipedia.org/wiki/2019_English_Channel_Piper_PA-...