A true Exorcism involves a nominated Priest that has been given permission by the Church Authorities to perform the Exorcism.
Strict guidelines must be abided by before such permission for an Exorcism is granted. The subject is studied to make sure that it is, as far as can be assessed, a true case of possession, because to attempt an Exorcism on a person who is mentally ill or just not possessed at all is dangerous for the subject concerned.
The Priest will be selected after careful consideration and will usually have an assistant. The Exorcism itself will usually involve the Priest, the Assistant, a member of the subject's family and often a medical doctor.
The Exorcism Rites are long and the Exorcism itself will often be extremely wearing on all concerned. To witness such an event would not be advisable, especially if someone in the room were to have a potential mental illness, in fact, it would be downright dangerous.
Exorcisms are not something to be taken lightly and contrary to some "exorcist's" claims, true exorcisms do not happen every day and neither do they consist of a mere blessing - a blessing alone will not expel a dangerous and determined spirit.
The subject will not be the one to usually request an Exorcism, it often requires a friend or member of their family to ask for help as the subject is usually already "too far gone" at such a point and only an outsider will usually have the means to ask.
The studies done by the Church before permission for an Exorcism is granted is usually along the lines of the things happening around the subject - repulsion at any holy objects, crosses, holy water - whether these objects are in view or hidden, but in the vicinity of the subject - a truly possessed person will know. Absolute detestation of the word "Jesus" or "Christ" - in fact they will often be completely unable to utter those names. Speaking in unknown languages - not gibberish but real foreign languages to the subject. Body contortions, explicit language and often poltergeist activity will accompany such a person. They will often have a truly horrendous smell around their personage and they may also know things about others that it would be impossible for them to know in normal circumstances. They will also find it nigh-on impossible to enter any Church or blessed place.
I hope some of this helps