ok, I'd like to preface what I say here with the comment that alot of this is my opinion, some of it is directly biblical, some of it is drawn from apocrypha etc. I don't claim all of this to be doctrine, its just my view based on what I've seen to this point.
ok, first of all, there is relatively little said in scripture about demons, or casting out demons etc. So little in fact that there is still debate about what demons even are. There are two common views within christianity as to what demons are, and a couple of less common views as well. The most common view is probably simply that demons are fallen angels. However, Angels are also a fairly mysterious topic and not much is said about them in the bible.
Some people do not believe that demons are fallen angels because of the fact that the few refrences in the scriptures to demons and fallen angels differ significantly. fallen Angels are almost always refrenced in terms of residing in the air above the earth, while demons are exclusively mentioned as earth bound spirits that walk the surface of the earth. Fallen angels are depicted as princes and powers while demons are much lower, begging and whining etc. another major difference is that demons seem to crave to inhabit bodies, while there is no mention of this at all for fallen angels.
In the bible Jesus does give some indication that different things are required to cast out different demons.. some times simply a command in the name of Jesus was enough, another time this was not enough, and Jesus said that that particular kind of demon could not be cast out unless the person who was casting him out had prayed and fasted before hand.
In the book of Tobit, which is part of the apocrypha, there is an instance in which the angel Raphael shows a person how to drive a demon out of a home by burning a particular substance. Granted the apocrypha may not be inspired scripture but it has been recognised by both Jews and christians for thousands of years as truthful and worthy of consideration. This could indicate that there is much more involved in the spiritual world than we think.
An idea often put forward is that non christians casting out demons is basicly a trick to decieve. However, this accusation was leveled against Jesus and his response was that Satan does not cast out satan, and a house divided can't stand. This would tend to indicate that demons do not cast out other demons, or satanic power can not be used to cast out a demon. Thus if a demonic manifestation is cast out, and I have seen instances where it seems to have been, I don't think "they're doing it by occult power" is a valid answer.
One thing to consider of course, is that Jesus said, when a demon is cast out, it wanders for a time, and then returns, and if it finds its former host empty and unprotected, it will go in and bring friends this time, and make things worse. Even if non christians are able to cast out demons, they are only making things worse in the end if the victim doesn't turn to God and the protection of his Spirit.
