Viruses "breed" (replicate) within infected cells, not within masks. Other kinds of microbes such as bacteria could breed within masks provided they find substances they can feed on in this environment.
Sars-Cov-2 has been shown to survive for extended periods of time (hours) on various types of surfaces. There is no reason why masks would be different. The following study found that Sars-Cov-2 remains viable up to 72 hours on plastic. Surgical masks are typically made of polypropylene or other polymers.
However, a more recent study found that Sars-Cov-2 is unlikely to remain viable for extended periods of time on porous material such as fabric.
Also to this day the following study appears to be the only one documenting a case of indirect contamination.
(Obviously, I cannot prove the non-existence of other such studies.)
As a result, droplets and aerosols (smaller droplets that remain suspended in the air) are thought to be the main vectors of contamination. Secondary contamination due to contact with a contaminated surface is thought to be a rare occurrence.
In the case of masks, one would probably have to touch a mask which has been recently worn and not wash their hands afterwards in order to be contaminated. It seems unlikely to me that this would be vector of transmission.