B737-800 First Officer reported the fire extinguisher in the flight deck began leaking during cruise. The flight crew donned oxygen masks until the leaking stopped; and then continued to the destination airport.

Narrative

While in cruise; we heard a popping sound followed by a short hissing sound. To me it sounded like an O2 mask or line broke and then was slowly leaking. We looked around the for the source and the Captain discovered it was the Halon Fire Extinguisher by feeling air from the pressure gauge. After a moments pause of realizing what happened; the Captain directed us to put on our O2 masks and establish communication. Once we completed that; the Captain transferred the controls to me and started looking for the correct QRH procedure. There were no direct correlation procedures for our situation and so we choose to start at Smoke and Fumes checklist as it was the highest level of danger we could start with. The Captain did a great job of keeping the situation calm and at the right level of urgency. He contacted [Operations] and was then connected to [Maintenance Control] as well. [Maintenance Control] did not have any direct guidance available for our given situation. Next the Captain ran the Toxic Substance check list. This did not really apply because it was directed more towards a cabin problem. During the whole situation; the Captain asked how I was feeling and if I felt any effects of possible halon inhalation. I never felt any ill effects. [Maintenance Control] was unable to give us any good information on how long the Halon would last in the flight deck due to no procedures for this event. After about 20 minutes we felt it was safe to remove our O2 masks and continue to ZZZ. We made that decision based on not seeing any evidence of Halon discharge (we later discovered we were both mistaken on what that would look like); the discharge from the fire bottle seemed to have stopped (no sounds; no felt air coming from the previous spot and bottle pressure holding steady for the last 15 minutes); and after some additional information from [Maintenance Control]; we felt the air had sufficiently circulated through the flight deck. We both kept our O2 mask at hand and ready to don again if the bottle where to release again.The location of the fire extinguisher lends itself in inadvertent contact as crew and team members transit the flight deck. Meaning; as people get out of the left seat; they can be making contact with the bottle without noticing or as a way to stabilize them themselves if they lose balance getting out of the seat or the jumpseat. Over time it is conceivable that the gauge could lose structural integrity to the point of not being able to hold the correct pressure any longer. Suggestions to Prevent: none that I can think of now. It would have been helpful to have known what to expect when the bottle is discharged (i.e. powder versus clear gas; odor; effects; timing of effects.) A quick brief on what Halon looks like; how it effects the body and perhaps a review on if a QRH procedure might be needed might be a few ways to mitigate problems if this were to happen again.