Sorry. This turned into a bit of an essay.
I think any career hotelier has suffered bedbugs. Especially one working 3-4 years ago, when there was a global problem. A law had been passed and quite rightly so, that the pesticide used to treat the issue was banned as it was stupidly toxic beyond any way that could be permitted. Here lay the problem; no one pesticide could completely solve the issue. A combination had to be used to eradicate the little gits. In the period it took to get everything back to normal there was a global outbreak, even making it into the global press. I believe one of New York's most prominent department stores had to close for a number of days, there were reports of people getting them from public transport (although I don't know how entirely true the latter is).
I think in a year period I had to have an exterminator round for ten different outbreaks. I should point out to people who are fortunate enough not to have had an infestation and that's what it becomes in an incredibly short space of time, it has nothing to do with the cleanliness of your hostel. A traveller basically carries their travel stay history in their bag. If they've stayed somewhere that had them they'll crawl into their bag and lay egg and go with them on their next journey. Sadly this meant for a lot of people they went on the holiday of a life time, had it spoilt (if you've ever been bitten bad you know that spoilt is not an exaggeration) and finally got home to escape the nightmare only to discover said nightmare had come home with them and their house was infested. They're really that prevalent.
Any claim of bedbugs must be taken very seriously (although there's enough idiots out there who cry bedbug because of a mosquito bite and refuse to believe otherwise). What I will do is first inspect the room. There are some good bedbug traps out there that are quite cheap. Their function is not to get rid of the problem rather than to let you know if there're bugs in there the first place. It's also worth checking the seams of the mattresses and other nooks and crannies. Alternatively it's pretty obvious you have a problem when someone is covered head to toe in hundreds of horrid red bites each one hurting like a bee sting. Bedbugs don’t pop out of a handy human snack before toddling off to bed they like to graze. You can see their route in a path of bits 5mm-10mm apart. I’ve had suffered a really bad attack and you just feel violated. I must have had a good five hundred bites and you hurt everywhere because they bite everywhere and I mean everywhere! Head, face, limbs torso, arse, genitalia. Everywhere!
A mistake that I know a lot of people make is to treat just the room. I think as hoteliers we have a duty to make sure that the infection does follow the guest on to another hostel or back to their house.
There's two options here. Treat the room yourself (better for the more experienced hotelier) or get an exterminator in. If you go for the latter they generally have a contact poison that mimics the pheromone that draws them out of hiding. If this is a private room and you have the fabric covered box sort of bed they will have to rip open the bottom and treat inside. The room then cannot be entered into for 24 hours.
If you are doing this yourself it's better to get a complete bug kit. You'll get smokers (smoke bombs that fume the room), sprays and powders. The sprays should be a pressure pump bottle with spray hose (like you get for weeding). This you spray everything bar the electric appliances and the wall sockets. Walls, ceiling, cracks (especially), under tables. Everywhere. Then you use the powders, these are used on/in the electric sockets. Lastly you close all windows, light the smokers and leave the room for a good four hours (obviously on at a place where you’re not going to cause a fire. The floor is best). Personally I will throw out all bedding. It's expensive but not as expensive as if it doesn’t work and you have to treat the room again plus any room that they’ve managed to move to and they do so very fast. You can keep the bed linen if you must but it has to be machine washed at over 60c. This will kill both bugs and eggs.
With the guest it is important to emphatic, caring and extremely helpful but make them understand that this is not the hostel’s fault. The bugs did not originate at the hostel but most likely would've been from the previous guest or possibly even themselves (if they only have a few bites). I always make it policy that I tell them that we will wash everything for them at our laundry but they have to be done at over 60c to get rid of contamination. If they have anything delicate they'll have to either risk it or pay for it to be dry cleaned (also works). I'll also tell them to leave their empty bags in the room to be treated along with everything else. Until the guest’s luggage has been treated I won’t let them into another room. That room needs to be kept a keen eye on for the next week or so. Obviously in this time they needed to be treated like royalty. I’ll also go out to the pharmacy and get something like calamine lotion or an aloe gel to help with the bites.
If handled well the best you can hope for is no further re-emergence of the incident and a guest that doesn’t leave a review. Interestingly bedbugs became almost extinct in the 1990’s are re-emerged with a vengeance partly because of the advent of cheap air travel and party because more people were heating their homes continually throughout the cold months rather than wrapping up warm when indoors.