Can I transferring from private to public university?

Hi Matteo,
In short: For most study programs, you shouldn't encounter major issues when transferring to a public university.
Just ensure that your Bachelor's degree program will provide enough ECTS for the corresponding Master's program. Also, check whether there are different language requirements and if your initial degree is in a suitable subject.
However, be aware that the capacity for Master's programs at public universities might be limited, and there could be a few exceptions in compatibility for some programs. For instance, a B.Sc. in Psychology at a private UAS might differ from the state universities' Psychology program in terms of state examination and prospects for approbation. (https://www.privathochschulen.net/en/courses/social-sciences/psychology)
Kind regards