We're talking about a collaboration tool here, and there is no jurisdiction that I know of where employing such tools, even when mandated by the employer, is unlawful; definitely not related to GDPR which is completely out of scope here.

Of course, as any tool can be used for bad things, so if, say, instead of the default sharing of just the development apps, the employee shares his browser, email client or instant messenger which he uses for personal purposes, you could argue it crosses the line into unlawful workplace surveillance, so it becomes a matter of setting correct policies. Sounds to me like an enterprise feature set you could charge for, as complement to the free tier.

As soon as private information (e.g. an email) is opened on that machine, you're in violation of workers rights and gdpr.

This is a wild exaggeration. Again, GDPR is not relevant here and private communication on the employer's infrastructure generally caries no reasonable expectation of privacy, at least in the US, double more so when you are actively sharing that email with your team.