It depends. If a foreigner is exempt from the need to have a work permit (e.g. a graduate of a Polish university), he or she may legally work for another employer, without the need for a work permit. This is due to the fact that he/she is legally entitled to work without a permit. Moreover, the decision on the basis of which a temporary residence and work permit is granted to a foreigner who is a graduate of a Polish university (who joined the full-time graduation diploma in Poland during the procedure) does not specify the employer. It is enough for this foreigner to provide the new employer with a copy of the full-time graduation diploma in Poland. This copy must be included in the employee documentation in the case of an inspection of the legality of employment carried out, for example, by the Border Guard.
However, it is not possible to admit work to a foreigner who is not exempt from the need to have a work permit. In this situation, it will be necessary to register a declaration on entrusting work to a foreigner or submit an application for a type A work permit for a given foreigner.