Ill answer this for the FAA (here in the US) but keep in mind this differs around the globe.
The FAA outlines some of their noise abatement information here. They created a special website that has a lot of the data here if you are interested.
There are 2 basic scenarios here in the US, towered and un-towered operations. There can be a noise abatement or related procedure in place in both cases and it is both the pilots and towers (when applicable) responsibility for knowing them and abiding by them. Some of them may be set in place by local governments (if its say a municipal airport) and you may face local legal action for violating them.
At a towered airport that has a noise abatement the tower will generally not vector you to that area or have you use a given runway provide conditions allow. For example when I fly out of KPNE there is an abatement in place, in this case they only list Rwy 15 arrival (stay above the glide slope) however there is another abatement in place on the weekends to keep you from flying over the heavily residential areas early in the morning. The tower vectors you appropriately for that one.
At un-towered airports there may be a published noise abatement that you are requested to follow. For example I fly out of KDYL which has a noise abatement in place (to keep you from flying over downtown). In this case we can see it noted in the AFD. Although its not fully explained the procedures are for RW 23 turn to 250 as soon as practically possible, for RW05 no turns under 1200 ft. and there are no touch and go's at any time.
In both cases the procedures have a similar outcome to what you have described, keep the airplanes away form the populated areas. The procedures vary heavily per airport and scenario but all generally strive for the same outcome.
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). 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
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. Over all the answer to your question is, yes airports can fine pilots (well really owner/operators) for noise violations.
There are also noise limits on aircraft that deal with engine noise but are generally handled during the certification phase of the air frame. This helps to reduce airport noise overall.
As for global regulations the IATA has outlined some stuff here with the full text here. ICAO has a set of policies you can find some references to here.