This actually a confusing area of the law that varies greatly from state to state. Therefore, I would recommend that you ask a local real estate attorney. You may also want to check with the insurance broker who procures your homeowners insurance and ask if you need any special coverage due to the existence of the easement.
As a general proposition, a landowner is always liable for an injury to a dangerous or defective condition that exists on his or her land. When there is an easement, both the landowner or the easement owner can both be held liable. Often the document creating the easement, my contain an indemnity clause stating that if the easement owner or his invitees are injured while utilizing the easement, then the easement owner will defend and indemnify the landowner in any lawsuits. Therefore, you should read the original easement to see if such a clause exists.
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