In a recent survey conducted by the group, 92 per cent of councils described Royal Mail as 'inflexible' or 'very inflexible' when it came to meeting their requirements.
Only 13.5 per cent of councils said they were satisfied with the service they received, while 38 per cent were 'actively dissatisfied'.
Nearly 65 per cent of users said they often complained to Royal Mail about aspects of its distribution service, while 90 per cent said the organisation did not handle their complaints well.
Gaudin said: 'It has been the case to date that many councils have had little or no choice of provider for regular newsletter distribution, but this is changing. Royal Mail needs to up its game or accept it will lose out.'
Councils have a legal duty to communicate with residents, but Royal Mail is limited in the number of commercial postal bookings it can take.
Council boundaries do not follow postcodes, so people living under one authority may receive information from a neighbouring council, said LGComms general secretary Kevin Wilson.
Royal Mail has pledged to resolve these issues with a deal to be announced at LGComms' annual conference on 13-15 April.
Royal Mail's monopoly of the postal market will end next year when it is opened up to competition.