If you've taken a hard look at your secret squirrel company stuff and you've decided that 1) these ideas truly are trade secrets worth stealing and 2) they need to be documented in detail inside the product, then yeah, you might want to look at some kind of self-hosted option.
I suspect however that 90% of these concerns are simple hubris, much akin to "sign this NDA before we talk about my startup idea."
If my clients knew I was telling everyone that they run my software, I'd be sunk. And probably sued.
If they sold the software maybe you could host it on-site, but having this on centralized servers with many other companies is incredibly risky and generates a huge target.
An effort to break into a large cloud company storing thousands of interesting accounts has a bigger payout. Attempts to attack such a service will be made more often and with more sophistication than attacks on a smaller company only having one set of potentially interesting secrets.
If your company really values its corporate info so much, it likely already has policies and expertise to secure its own servers. If not, maybe a cloud instance is safer.