My current position is at a large-ish company that has a well-defined process for laying off or re-allocating employees when a project is terminated. This process is called "reassignment" and it works like this: You (the employee) is relieved from your normal work duties for a period of time, a couple of months typically. During this reassignment period, your work job is to find another job, hopefully with this company but also maybe external depending on everyones needs. If you haven't found another internal position at the end of reassignment, then you are let go with a small severance pay.
It looks like my new job will be 100% remote, but I have several weeks left on my reassignment before I get laid off and then get my severance. I have transitioned all of my responsibilities at the old job and they do not expect me to come in the office. I don't want to resign early (lose the severance pay) but it's frustrating to just cool my heels when I could start at my new position already.
These companies are not competing and I don't believe there is a conflict of interest between them.
What problems am I inviting if I start my new job before being laid off my old one? Legal problems or will it damage my professional reputation? Who does my current employer report my start and end date to and are they allowed to share that information with just anyone?