You can be responsible for the way water moves through your property and this sounds like a water egress problem from your land to theirs. Are you sure that it is a "stormwater" easement and not a drainage easement? Generally speaking, the person owns the land that the drain goes through, but the drain/pipes/etc belong to someone else (I would try contacting the water authority). The easement is simply a right to use the land. The next question is 'who has that right'? It may be the water authority/council, or it may be another neighbor so that they have access to connected pipes. A title search of the property may also assist i discovering who owns the pipes.