Lots of good answers already but I think there are still some important things to point out. This is one of the hardest challenges for a parent.
What type of drugs are we talking about?
There are three groups of illegal drugs that are worth differentiating:
A) Alcohol, B) Weed, Hashish & Marijuana, C) prescription drugs and hard drugs. All three of those are illegal (depending on age and country) but I think they need different approaches. For now I'll assume this is restricted to alcohol and/or weed. If you suspect it's something worse, please seek professional help ASAP.
Is my kid taking illegal drugs?
Of course he/she does at least at some point!! According to www.drugabuse.gov (US centric) 40% of all high schoolers got drunk 17% smoked weed just in the last month. The number of high school students that actually graduate without any exposure to alcohol or weed is vanishingly small. I know that all of my kids had exposure before graduating. I grew up in a different country a long time ago, but even then the picture was the same: Access to alcohol and weed was always there and most kids participated to some extend. But that's not a necessarily a bad thing:
What is your goal?
Before you dive in here, please be clear about what you are trying to achieve. In an ideal world you child would never be taking any illegal (or legal) drugs for their entire live but a) this is not going to happen, and b) it's not your choice anyway and you have little or no control once they leave home. A better and more realistic goal could be the following: "teach my kid about the risks and rewards of drugs and enable them to make responsible decisions" . This is very different from "drugs are bad, don't take them".
In the end you want your child to become a responsible adult. Most responsible adults will occasionally engage in some sort of drug use (illegal or not), so they need to learn the tools of how to do this.
How do you do that?
Keep the communication lines open and keep judgement to a minimum. Be clear about the message but be realistic, honest, factual and by all means avoid hypocrisy. Don't tell your kids that alcohol is a terrible thing when you pour yourself a drink first thing you come home from work. Don't lie or be overly dramatic. A statement like "you cannot smoke weed because it leads to addiction on hard drugs" is simply not true and your kids know that already. Speak openly about what you know and what you don't know, do some research together if the kid is up to it, but be open enough to accept the results of your research.
Set some "common sense" rules: MIT for example allows alcohol in dorms but always a requires a "party monitor": a person that is sober and pays attention and can intervene when things get out of hand. In general "always have a person around that is clear headed and that you trust".
Allow for some "training": In my opinion it's preferably for kids to have their first exposure at home in a safe environment than to wait until they are for example in college where things are much more dangerous. At some point they need to figure out what e.g. alcohol does (or doesn't) do to them and how they react to it, so it might as well happen when there is a safety net available.
Many drugs are indeed illegal which means that you and your kid can get in trouble with the law. You and your child should be aware of the actual regulations and the potential consequences. In Massachusetts for example, weed is actually less of a problem than alcohol. Weed under 1oz results in a citation and $100 fine. Alcohol, however, gets you often (but not always) in front of a judge with a significant chance of a criminal record. Again, it's important to do the research and stick with the facts and accept them, even if you personally feel otherwise. Do this with your kid together. Make sure he/she understands what the laws and the potential consequences are. Factually, not threatening.
Should you search your teenager's room?
Heck yeah!!!. Here is why (which you should clearly explain to them): Depending on your local legislation even parents & siblings are at significant risk of legal exposure. If illegal drugs are found on your property, you may be legally responsible for this. If you end up in front of a judge and end up with fines, jail, or get a criminal record the whole family suffers. Most teenagers are NOT aware that storing some weed in their room or raiding the liquor cabinet with their buddies can result in serious legal consequences for the parents. You need to be very clear about this: "Mike, we cannot have any weed in the house because if this gets out, I'm legally responsible and since my company has strong no-drug policy I will likely get fired. I really need to make sure". The teen is going to be annoyed, but they tend to be permanently annoyed anyway so what's the difference :-). Searches in my house were actually productive, did not harm the relationship long term and actually triggered some good discussion.