Aircraft are weighed by the manufacturer and then when introduced into service by an operator. Small operators are required to weigh the aircraft every 4 years, large operators of fleets are mandated to weight a number of like aircraft by a formula dependent upon how like many aircraft they have in the fleet. Essentially some of these aircraft will be weighed every 4 years but should not exceed 9 years with the aircraft that have the longest period between weighings being the obvious choice to be weighed next.
When aircraft go through a major service or have major component changes or modifications then they must be reweighed. If a small modification or addition/subtraction is made then this can be recorded in the weighing schedule of the technical log and a running total of the mass and balance kept. All additional role equipment etc will be added to the basic empty mass.
Large companies with their own engineering departments have the facility to weigh aircraft. Other maintenance organisations have the same facility. There are a number of specialist organisations around that offer a weighing service.
Search on you tube " how to weigh a 737 in 60 seconds".
Hope this answers your question.