I have a monthly recurring charge for a service I receive. Last month,
the service provider charged me twice. The second charge caused an
overdraft of my bank account.

The bank says they can't refund the overdraft and I figure because the
service provider screwed up, they should refund the money.

Do either of them legally owe me the money or am I out $30?

Contact customer service at the service provider.  Bug them until they
pay you the $30.  The mistake was theirs.  Call, write letters, demand
to speak to supervisors, manager, vice presidents, etc.  Write to the
president and CEO if need be.  Persistence pays off.

It probably depends on the state you live in, but in most cases you
are ultimately responsible.  Now as good customer service, the company
that double billed you should refund your $30 to you, but I honestly
do not believe they are obligated to.  If they refuse to cover your
$30 charge, you could probably take them to small claims court, but
there is no gaurantee that you would win, and most likely, you will
spend more time and effort fighting it with them than it is worth.  I
would chaulk this one up to one of life's experiences, and if they
double bill you again, cancel the automatic debit from your account,
and setup an on-line bill payment with your bank to just send them
their money each month.

Personally I prefer the online bill payment method better, because
that way if there is a double payment, it would be my bank's fault,
and their wouldn't be a problem with them crediting any bank fees
"their" mistake generated.