What would be the potential consequences of offering to fund your own graduate degree? I am aware that being funded by a grant greatly improves your chances of receiving a master's degree, however, hypothetically what would be the potential consequences of offering to fund your own? If the negative consequences are minimal, how does one go about offering this to a potential supervisor without under-selling themselves?

I'm applying to programs in Canada.
Getting funding usually stipulates that you have a project that is related to the fund that you have applied to. That being said, it is not always the case. Whether you are self-funded or funded does not necessarily mean that a particular supervisor will work with you. If your research interests are not aligned with the professor's, then it is likely that he/she will reject working with you, even if you are funded. It does not really matter if you are funded or not. It looks better on your CV, however, if you do get funding.

Furthermore, being funded does not guarantee that you will receive a master's degree. Hard work and dedication do. It is the same in any academic program.

If you want to work with a certain professor, I would suggest that you email him/her with a research proposal. That way he/she is able to 1) know that you exist and are keen to work with him/her, 2) that you have a project in mind, and 3) you are open to criticism concerning your research proposal. 

Professors like working with students who want to work with them, usually, and they also like students who have similar research interests, or have stimulating proposals for new avenues of research.