I'll answer the question more of who “does” right now. The “should” is maybe a bit trickier.
In other jurisdictions, such as the United States, this is a power generally confined to governments, the federal government in the U.S. and state governments. It is a bit broader in the U.S. in that state tribes are also authorized to sue. The reference under the American legislation, under CERCLA and OPA, is to trustees, that being the federal government, state governments, and state tribes.
I have in the past, blogging about Bill C-22, or ESSA, suggested that there might be scope here to broaden the category to recognize aboriginal governments, Indian bands and such, to claim for such damages within their territory. You could expand it to include municipalities. With the disaster that happened at Lac-Mégantic, amongst the tragic loss of life was also a massive environmental catastrophe. It seems to me that the municipality there should be empowered as well, frankly, as a representative of the people.
I guess my bottom line, to try to keep it simple, is that governments, various levels of governments, generally are accepted as being the right parties to sue for such damages.