Postdoc: established project vs your own
Your second option many not even have a &quot;supervisor&quot; as such. It might just be an opportunity for an aspiring academic to work with a small research group on questions of mutual interest. This might be especially true in mathematics, for example. It is a way to get some experience, possibly even including teaching a bit.
I've been in a department where a post doc was hired specifically to work with a junior faculty member to boost their mutual CVs and experience.
Another situation involved a person who had an interesting (to some of the other faculty) research program so they hired him to work with the others, bringing new ideas into a research group. It was mutually beneficial as was my other example. In this case the long term intention was to offer the postdoc a permanent position later if possible.
There are hardly any disadvantages as long as the postdoc works well with others as a colleague and both gives and accepts help.