If you assume simply generally sloppy/user-unfriendly data access: you're fine. Windows can't do anything (e.g. access flash drives or network data) while it's not running, and doesn't know how to parse Linux file systems so it can't read your Ubuntu drive (unless you for some reason formatted it with NTFS, UDF, or a FAT variant rather than Linux defaults like ext4 or btrfs) when it is running. To Windows, the other SSD is a black box of meaningless nonsense.
However, that won't necessarily stand up against an actively malicious piece of software. It's very unlikely that Microsoft starts shipping an ext or btrfs driver with Windows, but it's possible; NT drivers for such file systems do exist, or there's things like WSL (which is disabled by default still) which could mount the other volume if granted access. Similarly it's extremely unlikely that any not-specifically-targeting-you piece of Windows malware is going to bother to include support for Linux file systems, so the other SSD will be meaningless nonsense to them too. But it's not impossible.
Similarly, it's not impossible for Windows (or highly-privileged and advanced malware running on it) to bypass nearly any protection you could put in place. For example, it could easily lift your entire Linux machine into a transparent VM, such that even when you think you've shut down Windows and rebooted into Linux, it's actually still running there, with a lower-level access, able to watch everything. Similarly, as another user mentioned, it might be able to install a keylogger into the machine firmware and can definitely overwrite the Linux bootloader, to introduce a keylogger that captures your disk encryption password. The only real protection is to use a totally separate machine.
On the other hand, you really don't need to be that paranoid about this. Microsoft has no interest in monitoring what you do in Linux; it has reasonably-legit reasons for the monitoring it does in Windows (most of which can be turned off) that mostly wouldn't even apply to Linux and certainly wouldn't be worth the development effort. Not to mention the PR shitstorm when it eventually came out. Windows might be closed source, but that doesn't mean it's an unknowable block box of mystery; there are almost certainly more people monitoring every network packet their Windows machine sends or receives than there are who have read more than 0.1% of the Ubuntu source code, not to mention the thousands of Microsoft employees who work on Windows and related stuff (meaning they do get to see the source code) but do in fact have ethics and sufficiently well-padded bank accounts that most would be willing to object internally to such overreach (which might make the news) and some who would be willing to publicly blow the whistle on it (which would definitely make the news).
