Could you clarify where the privacy concern is here? As I understand it, I'm sharing my IP with search engines anyways; the only difference with a self-hosted SearX instance is that I'm sharing my server's IP instead.
Is the concern that the latter's IP isn't behind a NAT, and therefore is more unique? If so, I think that's the least concerning of the identifying datapoints that a search engine has access to -- my browser metadata is far more identifying. With SearX, that information doesn't get forwarded (IIUC).
Server IP can easily be differentiated from casual internet users' IP, and since you also don't usually get a server IP rotated, you're pretty much stuck on a single IP and you're easy to be categorized as an unique user no matter which device or location you use it from.
Is the concern that the latter's IP isn't behind a NAT, and therefore is more unique? If so, I think that's the least concerning of the identifying datapoints that a search engine has access to -- my browser metadata is far more identifying. With SearX, that information doesn't get forwarded (IIUC).