Way back when fire departments in some cites were run on a subscription basis. You contracted with the private fire service for a set amount of time. They then gave you a plaque that you mounted on the gable end of your house. The idea was that if the house caught fire people would call that fire service who would come out and put out the fire.
many districts had multiple fire services. Each catering to its own customers. Often, in part because there were multiple services, multiple companies would be working on one fire. Each only concerned with fighting the fire that threatened the buildings they had a contract with. Fist fights over who held the contract, who would get the glory of rescuing endangered people, and how to fight the fire were pretty common.
In theory all firemen had a duty to rescue trapped people. This was a lot easier to do in those days because fires were mostly wood burning and not very toxic. This was well before plastics and the deadly fumes they produce. There were no respirators.
These were the iconic days for firemen. Firetrucks were pulled by horses. Dalmatians really did ride on the trucks and pumpers were either pumped by hand or by steam engine. Firehouses were a mix of mens club, quasi-military order, fraternal order and outlet for young mens heroic impulses. There was a lot of competition for the glory of rescuing people and firemen were widely revered as heroes.
Problem was the system was not efficient. There was a lot of duplication of effort in rich districts. While the poor districts were neglected. Fires would sometimes start in buildings without contracts and grow too big to put out by the time it clearly threatened contractually protected buildings. Sometimes people died in houses while firemen fought over who would get the glory of rescuing them.
Also there was the small matter of fire companies getting their contracts based on reputation and threat. Competing fire companies sometimes resorted to setting fires to embarrass the competition. Some companies operated what amounted to a protection racket. People who didn't buy protection tended to see their houses burn.
In the end it became very clear that in some situations private, selective and for-profit services just aren't a good deal. Some problems are better handled by a more united collective effort. which is why the municipal fire service came into being.