Dear Thomas,  

You question is a good one, and it points to something fundamental that we can very easily overlook in a vocation. You see, we tend mostly to deal with a vocation from the point of view of our feelings, and this makes us think that if we feel we have one it means we do, and if we feel we don't it means we don't. So your question is very perceptive. Not everything depends on our feelings and our good feelings need to be tested to see if there really is a call there, as also our negative feelings (against a vocation, or giving up on a vocation) must be tested to see if they really mean there is no vocation there.  

The basics to a calling are: having the necessary human qualities (physical health, appropriate psychological maturity and self-mastery, the necessary intelligence); having the necessary spiritual qualities (appropriate stability in the life of grace, for example); and having the proper motivation, being interested in the priesthood for a spiritual reason.  

As regards the screening process, principally it will address the areas I have just mentioned. Each particular vocation (the diocesan priesthood, this or that religious order) will have its own standards or criteria, and some may have additional areas they base their screening on (for example, some groups will not accept a candidate who has not finished college). Generally speaking, they will want to get to know you personally, then they will ask you for documents on those areas that can be addressed through paperwork (transcripts, medical exams..); then they will have some system set up to address those areas that require specialized assessment (psychological testing, for example), they will usually have you do some live-in with the community so as to get to know you personally.  

Screening is most visible when you are asking for entry into a seminary, but it is not a one-time event: it will be ongoing during the formation years, and will generally center on your progress in the different areas of your life as a priest: your spiritual progress, your human maturity, your studies, your social and communication skills, your attitudes, your love for souls and the Church, etc... 

Hope this helps. God bless.