This is a highly complicated, debated and complex topic. Whenever talking about zen, zen priests and zen masters we discuss a topic that is very hard to explain because in different cultures we have different meanings that could appear. Generally speaking, the zen master is a person that managed to reach a certain level, one at which he can teach and practice at the same time.
The zen priest is normally a person that went to the temple, meditates and wears robes. There is no criteria that can be mentioned about this. The one real merit that appears in this case is normally commitment.
The term “ZEN MASTER” can mean so many different things. In most situations we refer to a teacher that now does not have a teacher. Zen is basically going to have an idea of transmission to the student from the teacher, in a direct way, all going towards connection with Buddha. Many argue that this is incorrect and that this does not exist but that is what we see in this religion. The main idea is that the teacher is going to teach someone until nothing can be taught and the student can go on without needing help.
A zen master is basically an individual that is “authorized” by a teacher, one that was “authorized” by his own teacher, to practice and teach without having to be supervised. Zen masters can thus open a temple without supervision. However, not all of them do that. Some do not want to teach as they do not feel ready.
Authentication should, in theory, imply enlightenment. This is where things get tricky. Not all those that promote themselves as being zen masters are actually legit. Some just want to take advantage of people. They will teach for money and do not really offer the spiritual guidance that the student would get from someone that is a true zen master. Because of this, it is really important that you are careful and that you only consider those zen masters that are legit and that can tell you about their teacher. When this is the case, you would receive a much better guidance.
Remember that in Buddhism the monks-nuns are required to take a vow for their entire life. This includes chastity, poverty and various other restrictions. Laity (commonly referred to as householders) imply such vows taken just for a short period of time. Priests are people that take vows seriously. Masters are priests that go to the next level so the vows should still exist.