I have worked for 40+ years with troubled teens, in both treatment and correctional settings. I have a two-part answer for you. Before I say anything else, I want you to understand that there are no -- repeat, no! -- answers to your problem that will fix it immediately.
First, there is one thing I have learned for sure. Do not give teenagers orders you cannot enforce. It makes you look weak (and maybe ridiculous); teenagers do not need someone in authority over them they perceive as weak. All that does is reinforce whatever inappropriate tendencies they already have -- and every teenager has them -- okay maybe only 90%.
You cannot in actuality absolutely make a teenager do anything, except for making them not live with you. You can't control for certain where else they might live, but you can insist, eventually, that they not be in your house. This would mean getting "the system" involved. While you can't put teens out on the street, you can file complaints with whatever authorities there may be where you live to deal with teens who refuse reasonable adult control.
If the authorities cannot help ameliorate the situation by going through a process of several steps, and if they don't perceive you and your wife as unfair and unreasonable, the chances are great that they would grant you relief.
Some places would tend to be more sympathetic with your problem than other places. Some social workers you come across always take the teen's side, and assume the problem is you. Others understand that an adult who is not a teen's parent will need help getting the teen to accept that you have legitimate, if limited, authority over their behavior when they are living in your residence, eating your food, and wearing clothes you paid for.
Ultimately the key is to present them with choices, with consequences you have control over. Not knowing the details of your situation, I have no idea what those consequences may be. The point is that you get to decide, with or without their consent, what you will tolerate. They can't control you any more than you can control them.
As an example, if you and your wife believe it is very important that they keep you informed of their whereabouts, you could explain that you need to know that, and if they repeatedly (more than a couple of times a week, say) fail to let you know where they are, then you will require they be in the house by 5:00 p.m. every day for the next 7 days, and not leave again until morning without your approval.
If they violate those conditions, impose the consequence. If they refuse the consequence, or in effect tell you to go to hell right from the jump, then you for sure need to explain the ultimate consequence.
I have found in practice that you must lay out for the teen in advance what you expect, and what the consequences are for failure to comply. They need to know that for them to continue to live with you it is absolutely necessary that they give you authority over them, since you cannot make them do anything but leave.
Do not argue with them whether those principles are right or wrong. That is a trap you cannot get out of. Calmly explain that if they want to live there, they don't get to make the rules. Their only choice is to accept that you make the rules in your house, that if they cannot accept that fact, they can go live somewhere else.
Also, be sure to explain the following:
- you do not expect 100% compliance,
- you accept that teens seem to have a need to test the boundaries,
- you know that sometimes they will break the rules and get away with it,
- but that if they get caught, acceptance of the consequences is not negotiable.
When you present a choice (preferably a binary one), you must be willing to live with their choice. It would probably be a good idea to explain from the start what your goal is where they are concerned, what you can provide for them if they meet your expectations, and what you can remove from their access if they do not meet them. You must lay out for them the ultimate consequence, which basically boils down to banishment.
If these were children you have had a long and intimate relationship with, where there are mutual feelings of love and respect, such hard-nosed tactics would not be what I recommend. But if you are dealing with recalcitrant teens who don't automatically love and respect you, you really have no choice but to be matter-of-fact and no-nonsense in your approach.
I don't disagree with anything Mr. Gumble said. All I'm trying to do is give you an option if they flat don't care what you say or what you think. If they do care, Mr. Gumble's advice is a valid approach.
My second part answer is basically what Mr. Gumble advised, but I want to add some specifics to it. You may or may not have the resources available in your community, and you may not have the financial wherewithal, either.
If my recommendation is one you can make work, I'd waste no time getting going with it.
Research what is available in your area for family therapy, done by fully qualified, trained, and educated family therapists. In your situation, and in my opinion, a form of family therapy called Structural Family Therapy would work best -- unless I have misread the situation and you have teens who are willing to work toward making the family function better.
If that is the case, it doesn't matter much what form of family therapy is used. But if the teens basically want you to leave them alone, let them do whatever they choose, and have you pay for it all, you need to bring in people who are not going to give the teens a shoulder to cry on. You need professionals who will hold to account the teens, you, your wife, and to some extent your four year-old. Your family system as a whole is the problem, and everyone is contributing to it. The teens may be the most obvious problem, but you are all involved in very tangled feedback loops. It takes an expert to untangle it all and come out with satisfaction on every hand.
If the teens refuse to attend family therapy at least for the first meeting, should you arrange for it, in my opinion you are at the final consequence already.
Finally, if you do obtain private professional help, forget everything I have said and follow their advice.