Really only if you are paranoid. It's incredibly unlikely that the labs are lying about not training on your data for the API plans that offer it. Breaking trust with outright lies would be catastrophic to any lab right now. Enterprise demands privacy, and the labs will be happy to accommodate (for the extra cost, of course).
No, it's incredibly unlikely that they aren't training on user data. It's billions of dollars worth of high quality tokens and preference that the frontier labs have access to, you think they would give that up for their reputation in the eyes of the enterprise market? LMAO. Every single frontier model is trained on torrented books, music, and movies.
I just know many people here complained about the very unclear way, google for example communicates what they use for training data and what plan to choose to opt out of everything, or if you (as a normal buisness) even can opt out. Given the whole volatile nature of this thing, I can imagine an easy "oops, we messed up" from google if it turns out they were in fact using allmost everything for training.
Second thing to consider is the whole geopolitical situation. I know companies in europe are really reluctant to give US companies access to their internal data.