North Caucasian Federal District
It is very useful to have at least a basic knowledge of Russian, especially taking into account all the militarised road checkpoints. Since hitchhiking was popular in Soviet culture all people will understand what you are doing (including officials) and drivers should not ask you for money.
All the republic border crossings along the federal highway are militarized and most of the cars have to stop for a check. Only the driver is asked for documents and normally there will be no questions asked for a passenger (unless you obviously look like a foreigner or an outsider). In addition to border crossings there are many DPS checkpoints (traffic police). All trucks need to pay bribes at each checkpoint (be it border or DPS) unless they drive empty. Being in a truck is the least problematic, no one asks questions to truck drivers because they pay bribes anyway. Drivers will not ask you to contribute to bribes since the bribe amount is clearly known upfront and they have it covered in their budgets.
Hitchhiking in Karachay-Cherkessia goes surprisingly well. People stop quite fast and it is possible to easily hitchhike even smaller roads or unpopular dirt roads. In the mountains in the south it is possible to hitch for a few km and get out to see a nice landmark along the road, and then hop on the road again for another few kilometres and venture into the woods again.
Hitchhiking is alright. In Nalchik ask for a bus going to a city ring to go in the direction of Cherkessk or Ossetia.
Republic of North Ossetia–Alania
It is illegal for foreigners to travel off the federal road and the road to South Ossetia without a SPECIAL permit (it is not the border area permit). You can arrange it but it takes quite some time to get it and I heard of no person managing to obtain one. Thus, hitchhiking off the federal road is very risky as being spotted by police can result in you being deported from Russia (check Lonely Planet forum to learn more about such cases). I travelled with my friend in a private car off the federal road around the republic since there are no checkpoints so we bore some slight risk of being captured. However, admittedly going to the dead city of Dargavs is much more risky and we did not do it (a possible checkpoint).
Republic of Ingushetia and Chechen Republic
People in those republics are much more religious than in others making hitchhiking a great experience - people are very helpful, honest and friendly. On the other hand, they drive a bit more crazy than elsewhere. No special permits needed unless you want to see border areas (in Ingushetia amazing mountains, dead cities and bastilles).
Republic of Dagestan
Believed to be the most dangerous place in the Caucasus because occasionally bomb attacks and assassination take place. Hitchhiking is very easy and nice, but be careful where you are going. It is advisable, if not necessary, to have at least a basic command of Russian. All the big cities should be safe, even though you will encounter heavily armed police posts on the road and at the entrance of every bigger town. You will probably be recognized as a traveller when passing a police post and will have to answer a lot of questions. However, the police is usually helpful and often finds a lift for you, after you have satisfied their curiosity. A few of the forests are known to be inhabited by fighters. If you are spotted by them you have a high risk of being killed (which happens to some of the hunters). Thus, learn where you can camp in the wild and which valley you are going to. Additionally, a great many people drink and drive so be careful and be prepared to be offered some alcohol while on the road. Many people can invite you as their guests too. Use your gut feeling whether to accept it or not since some invitations can prove to be quite problematic (excessive drinking and lack of freedom to do your thing). It is easy to hitchhike at night too.
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