Bearshare has had a very storied history since the Supreme Court's landmark decision in Grokster II. That being said, one of the things that it does to verify the content shared between users (and it is the content that really dictates whether there is something illegal going on) is "acoustical fingerprinting" to prevent users from sharing pirated tracks. But in fairness, if there is a top 40 song available from another user for "free" that should set off alarms on any P2P network.
I hope this helps.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.
Bearshare is one of a zillion "distributed computing networks" that permit people to exchange computer files [these networks are oftentimes derogatively called "peer to peer networks"].
Some of the files being made available through Bearshare by its users have been unlawfully reproduced and are unlawfully being made available [many of which are sound recordings or movies, for example]. Some of the files being made available were, however, lawfully reproduced and are lawfully being made available. The percentage of the latter is, admittedly, small.
So ... if you download any of the millions of public domain works that are available via Bearshare then you have not violated the law. But if you download any of the millions of works that are not yet in the public domain then you have violated the law.
The issue is what YOU do, not what Bearshare does.