I've been using my furnace for 19 years without issue. Living room side cranks out the heat, dining room side has no flame of any kind. How would move forward to fix?
Call the Gas Company and tell them you are worried because your furnace does not seem to be working correctly. They will send a technician to look at it- FOR FREE. They want your appliances to work safely and correctly because they want to sell you as much gas as they can.
If you want to troubleshoot it yourself, you need to understand how they work. There are entire books devoted to this but here goes:
So I am sure there are differences between vintages and brands, but if you are familiar with millivolt pilot and burner systems you should be able to figure this out.
These furnaces generally have 2 separate burners. A single pilot assembly mounted at one of the burners is the ignition source for both burners, gas from the second burner (the one not next to the pilot assembly) is carried over to the pilot via a "crossover" channel, to allow it to also ignite.
A single gas control valve provides gas to both burners. The gas valve will remain totally closed until the control knob is set to "pilot" and pressed in, releasing a small amount of gas to just the pilot assembly, to allow you to light the pilot with a match (or piezoelectric igniter). The pilot flame warms up a millivolt generator (a cluster of bimetallic elements that generate an electric current when heated) called a pilot generator. Once the generator is sending current to the gas valve, the valve will remain slightly open to allow the pilot to stay lit when you cease depressing the control knob. If the pilot goes out or the pilot generator goes kaput, the gas valve will close and stay closed (unless you set it to "pilot" and hold the knob in, as described above) as a safety feature.
Normally, if one burner lights then both will light. Check the pilot flame, it should be a nice strong flame. The flame size can be adjusted with a special screw on the gas control valve.
Check the flame pattern and color on the burner that does light, it should NOT be a lazy, soft all-yellow flame. Nor should it be a hard all-blue flame; it should be somewhere between.
An easy initial action could be to replace the pilot generator, maybe its weak and not opening the gas valve all the way. Maybe you need a new gas valve? maybe there is debris obstructing the gas supply? maybe the "crossover channel" is obstucted, preventing burner ignition?