Either the first party (i.e. the client) runs the data on their own turf or they hand the data to someone else (you or whatever third-party you use) and trust t

Either the first party (i.e. the client) runs the data on their own turf or they hand the data to someone else (you or whatever third-party you use) and trust that the other end is going to treat your data right. I'm sure there's some sort of homomorphic encryption[0] magic scheme that might let you process the data on other servers or something, but I could not even begin to tell you how. Really, it's just trust.