While listening to this podcast about saving dying languages. A question came to my mind:
Does a deaf person from France understand a deaf person from Russia or any other country?
According the Ethnologue entry for Russian Sign Language:
Reported historical connections to sign languages in Austria and France, but not obvious from extensive wordlist comparison (Bickford 2005).
This implicitly suggests that they are not mutually intelligible (that is, that users of Russian Sign Language can't necessarily understand French Sign Language).
In addition, the entry for French Sign Language states the following:
Many sign languages have been derived from or influenced by LSF, but are not necessarily intelligible with it.
But before we stop here, I would like to talk a bit more about the logic behind this question. Perhaps your doubt is due to the following myth about sign languages: that the signs themselves have a direct connection to the referent. In other words, that signing is a form of mimicry.
In reality, sign languages behave in the same way as spoken languages: just like there is nothing in the word horse that necessarily connects it directly to the idea of horses, signs in sign languages do not necessarily have any connection to what they refer to. For this reason, there is a wide variety of sign languages as there is a variety of spoken languages.
Consider this mental exercise: If an American, a Briton and a Russian were asked to imitate the act of swimming through mimicry, it's likely that all of them would come up with their own way of doing it, but the mimicry would end up being similar because the act of swimming is the same for all of them. Now, if users of British Sign Language, American Sign Language and Russian Sign Language were asked translate swimming into their sign language, there is no guarantee that the sign for swimming will be similar across these languages. What's more, they wouldn't be able to come up with their own way to express swimming, they would have to use the sign that is present in the vocabulary of their sign language.
British Sign Language and American Sign language were invented separately, and they evolved before video was invented so there wasn't a lot of signed communication between people in distant lands. They have completely different grammar and mostly different signs. BSL and ASL speakers can't understand each other.
Auslan (from Australia) is a very distant cousin of Irish Sign Language, which in turn is related to British Sign Language. But an Auslan speaker and a BSL speaker would not easily understand each other.
There are even dying sign languages!
al-Sayyid Bedouin Sign Language is spoken only by one slightly inbred Bedouin tribe who, thanks to a genetic accident, have an unusually high incidence of deafness. But now the deaf al-Sayyid children are being sent away to schools that teach in Hebrew Sign Language, and their own language is in danger of disappearing.
I would like to add that there are even regional differences for one sign language. I know of someone who failed their BSL exam due to using some regional variations not native to the region she was taking the test in.
As a reason as to why BSL is different to ASL is that
In 1815, an American Protestant minister, Thomas Hopkins Gallaudet, travelled to Europe to research teaching of the deaf. He was rebuffed by both the Braidwood schools who refused to teach him their methods. Gallaudet then travelled to Paris and learned the educational methods of the French Royal Institution for the Deaf, a combination of Old French Sign Language and the signs developed by Abbé de l’Épée. As a consequence American Sign Language today has a 60% similarity to modern French Sign Language and is almost unintelligible to users of British Sign Language.