The biggest issue is that to challenge them, you have to dox yourself, even if the DMCA claim is completely bogus and itself is contains bullshit information about nobody who actually exists (e.g, in a doxing war, the party that fires a bullshit DMCA claim first has a huge advantage).
> Shouldn't offending side reimburse the expenses?
In an ideal world. They might even be legally liable in this one. But you still have to sue them to get the money, which is an expensive gamble for a very small pay off.