What I might say to someone starting to go this way.
On demon-possession and our authority
If my daughter's eyes were bugging out unnaturally and she were speaking awful things to me in an unearthly, gravelly voice, I'd start looking seriously into exorcism, and call around for advice. I don't rule demon possession out completely, without considering circumstances. I would not say we have no authority to exorcise. But I would not seek to take action on the spot. (Scriptural accounts imply long times of possession, not for an hour or so of disobedience). I would not trust to saying certain words to cast out a demon and restore my child to obedience (Scriptural accounts don't show possession as source of disobedience, but as tormenting the soul and body). I would not assume my daughter is possessed just because she is not being obedient, nor if she is physically defiant. It is not a normal part of the Christian's discipleship and fight against sin, to address and attack demons directly. This WAS more the case in the Gospels because of the unique circumstance of the Son of God being on earth. The epistles, especially Eph 6:10-20, give a strategy more addressed to us; the Gospels aren't giving us an attack plan but describing what happened to Jesus and what He had the 12 and 70 apostles do in that generation. Mark 16:17-18 is establishing the apostles' authority, and these things are recorded in Acts.
On linking charismatic gifts to salvation
Not having these gifts or powers personally doesn't question our salvation, undermine our faith, or take away spiritual warfare weapons. Many wind up believing their faith in God or standing in Christ is in doubt without these signs - that your faith is undermined if you don't think you can (or if you haven't) cast out demons, or spoken in tongues, or whatever. For the true signs of a Christian, see Rom 6:3-4; Gal 5:16-26; 1 John 3:18-23. Wonders, miracles, sensational Spirit gifts, and casting out demons are not among them. This is partly why I interpreted Mark 16:17ff above as I did.
So what do we do with reports of possession and exorcism today? I don't dismiss them all out of hand, but neither do I believe them all eagerly, to support the reality of the spiritual warfare we are actually in. I believe one's credibility is diminished greatly as he puts faith in saying certain words to automatically do spiritual good or evil, like with a deliverance prayer, or saying the name of Jesus. With that perspective, a report of demon-possession is less credible. Like speaking in tongues, it is more likely a result of the power of suggestion than a genuine demonic occurrence.
On agreeing to disagree:
This is where it gets hard. I actually think other Christians are seeing demons where there aren't any, and they are that deluded? Very possibly, yes. But don't take this personally as an attack on your rationality, if you believe demons have attacked you in ways you could sense with the senses. I just don't think the position above about how to fight them is the Biblical one, or that it makes rational sense. It is a difference of opinion that leads to not esteeming one's judgment as highly, as charismatics thus esteem my judgement less, in turn. We should amicably recognize that is a two-way street. But we remain brothers in Christ.