The FAA advises you to first report a violation to the airport but also provides contact info for directly reporting to the FAA. You can single out a given aircraft by accurately reporting the time of the incident (and if its visible) describing the aircraft. Granted it can be difficult at an un-towered field to know who exactly was departing if no one was around to see it or on the radio to hear it. Once reported the local authorities may have the right to fine you for the violation (if that is the punishment on the books). As far the airport them selves cant actually fine you as they don't have the authority to here in the US. For example in the NY/NJ area the Port Authority handles the violations and does hold the ability to issue fines.
Long beach airport lists their fines on their site
If a plane violates the Noise Ordinance, how much are they fined?
The initial phases of our violation process are educational: the first
time an owner/operator violates, they will receive a complete packet
including a copy of our Noise Ordinance, an outline of what the
infraction was, a letter from GANC offering assistance with
procedures, and a warning letter asking them to contact noise
abatement staff.
The second violation involves a notification letter
of infraction and a request for a written noise abatement program,
where we ask the owner/operator to communicate how they plan to
operate without violating our ordinance in the future.
The third violation is a fine of $100 dollars. The fourth and every subsequent fine of 300 dollars. Please note that Military aircraft
are exempt from this policy.
And it looks like, according to the regulations, the airport manager issues the fine and the payments go to the airport. This may vary by jurisdiction on how they chose to govern things like this.
