The trial period is a period of probation at the beginning of the employment relationship when both the employer and the employee can terminate the relationship without giving any reasons and without notice. It is usually between 1-3 months, but it cannot be longer than 3 months.
Although the residence permit of the employee depends on the job, you are still permitted to terminate the employment during the probation period, and the employee is still permitted to quit. However, in that case, they will have a very limited time to find a new job in Hungary if they do not want to go back to their country of origin.
The notice period is the period for which the employee remains in employment after the decision is made to terminate the employment relationship. Unless that happens during the trial period, the notice period is usually between 1-2 months, and whether the employee must continue working during this time depends on which party initiated the termination (the employee or the employer).
Learn more about trial and notice with a Hungarian work permit here.