Before starting, it is necessary to say that, in the case that a person believes to be possessed, he should first contact the bishop's office of his diocese. This can be done by telephone. There, they will set an appointment with the priest who is a specialist in these types of cases. If in the diocese there isn't anyone specialized in this matter, they will indicate which is the closest diocese which does have someone of this speciality.
We can say that some signs of possession are: entering into a trance with the eyes rolled back; sudden and irrational (due to its intensity) aversion to everything which is sacred: crucifixes, Mass, etc. Although this sign is not present in all possessed people, a characteristic which does appear always is a period of loss of consciousness in which a second malignant personality emerges. If all this happens after having taken part in some esoteric act, then there is sufficient reason to go and see the priest responsible for discerning these cases.
Appendix for priests
Given the pastoral and summarized character of the previous answer, it is not possible there to address completely such a complex subject as diabolic possessions. This subject is dealt with in greater depth in the rest of the sections of this web. The answer expressed above seeks to offer an idea to those who wish to ask a question about the matter. However, possessions are truly very complex phenomena. For this reason, it is not convenient that a priest make a discernment on his own account, but rather direct them to a specialist.
In spite of what has been mentioned above, it is a mistake to believe that possession only exists when holy water causes a burning sensation or that the person cannot enter a church or that one speaks in unknown languages. The most characteristic trait of a possession is that, having participated in an esoteric rite, the person suffers trances in which a second malignant personality emerges.
Many are the priests who think that possession is only something that appears in the movies. Although it is true that there have been some spectacular cases, it is much more common that the possession does not show other signs than those described in the majority of the cases which appear in the Gospel.