I have provided the relevant law concerning easements at the bottom of this answer. Parsing the "legaleze," the first question that you must resolve is: what does the easement grant actually state, and what would an objective person make of that grant?
If the easement is ambiguous, and states that the neighbor
has a "right-of-way," then the reasonable interpretation, in my view, is that the neighbor can make use of the road to access his or her property, and to provide access for any other persons who reasonable require access.
In your case, your other neighbor claims to be checking up on the easement holder's property. If that is true, then that is a reasonable use of the easement, and you cannot prevent the neighbor, and/or his dog from using the easement. But, if the dog or the neighbor strays from the road, or does not actually access the easement holder's property, then the neighbor is trespassing, because he is not using the road for the purpose for which the easement was originally granted.
If I were in your shoes, I would contract the easement holder in writing and ask if he has engaged the other neighbor to watch his property. If the response is 'no," then the other neighbor is a trespasser, plain and simple. Or, if you can video the dog off of the easement road, then that too is trespassing, because the easement is only the road, and nowhere else on your property.
One other issue is that Oregon law generally requires property owners to post signs at routine intervals notifying potential trespassers that they are crossing onto your land. So, unless the road is fenced in (even just a single wire fence), and no trespassing signs are posted, then you will have difficulty proving a civil trespass, except by accident, which is not actionable in court except for actual damages
Conversely, if you take all the precautions, and the neighbor is still tresspassing, then you can provide a written warning, and if the neighbor persists, you can call the police/sheriff and make a report -- or, sue the neighbor in small claims to discourage further trespassing.
Hope this helps.
[A]n easement holder can make only such use of an easement as is reasonably necessary to accomplish the purpose for which the easement is granted and the remaining dominion over the land upon which the easement lies continues with the servient landowner. Ericsson v. Braukman, 111 Or App 57, 824 P2d 1174, rev den 313 Or 210, 830 P2d 595 (1992).
"In construing an easement, our fundamental task is to discern the nature and scope of the easement's purpose and to give effect to that purpose in a practical manner." Watson v. Banducci, 158 Or App 223, 230, 973 P2d 395 (1999) (citing Bernards et ux. v. Link and Haynes, 199 Or 579, 593, 248 P2d 341 (1952), on reh'g 199 Or 579, 263 P2d 794 (1953)). In Tipperman v. Tsiatsos, 327 Or 539, 964 P2d 1015 (1998), the Supreme Court explained the legal principles that govern the proper construction
of an easement:
"First, in [easement] cases, 't is the duty of the court to declare the meaning of what is written in the instrument.' Minto v. Salem Water Etc. Co., 120 Or 202, 210, 250 P 722 (1926). Further, the court will look beyond the wording of the instrument 'only where there is an uncertainty or ambiguity.' Fendall v. Miller, 99 Or 610, 619, 196 P 381 (1921). If the wording at issue is uncertain or ambiguous, then the court must determine the intent of the original parties by examining the relevant surrounding circumstances. Doyle v. Gilbert, 255 Or 563, 566, 469 P2d 624 (1970) * * *. Such circumstances include the purpose and nature of the easement, the circumstances existing at the time of the grant or reservation, and the manner in which the easement was used by the original parties." Id. at 544-45.