Traditional search fund vs. self-funded search?

I have a quick question for search fund investors and would appreciate and your thoughts.

I am debating between raising a traditional search fund vs. doing a self-funded search. My question is - when a self-funded searcher brings you a deal for potential equity investment, are the deal terms typically the same as for a funded searcher? Or do the deal terms in some way reflect the fact that the searcher took more risk in undertaking the search at their own cost?

Specifically, would there be a difference in the economics to the searcher (e.g. higher initial earn-in, earn in preferred shares instead of common shares, lower IRR hurdle before the searcher earns up to his/her full amount etc.).

Or is it more common that self-funded searchers just look for smaller businesses that don't require much (if any) third-party equity capital (i.e. there is no difference in deal terms because self-funded searchers don't usually raise equity from others).

Thanks!