Based on relative length of its tail and also the relative lack of hair on its ears, I would identify this as a Rattus rattus, or ship rat (also black rat, and many other common names). We (OSU) had a webinar recording, Oh Rats! that would be helpful, but unfortunately it has been archived. I think this Marion County reference will be helpful, however: https://www.co.marion.or.us/HLT/PH/EHS/vector/Pages/rodentid.aspx#:~:text=Rattus%20norveglcus.&text=Description%3A%20Brownish%20gray%20above%3B%20grayish,Small%20eyes%20and%20hairy%20ears.
Rattus rattus are notable for their climbing abilities. They favor dense vegetation (or other masses of protective cover in sub/urban areas) and although the Norway ("brown") rat is well-known for its in-ground, burrows, the R. rattus is capable of burrowing. Because of their excellent climbing abilities, one must beware of continuing exclusion measures (e.g., blocking all vents with secure, unchewable welded wire) above the crawlspace and ground floor - thus the issue of "roof rats." 
Managing habitat quality such as abundance of food and safe denning and movement cover is more effective in the long run than ONLY trapping/baiting because of the reproductive and movement rates of rats.  (Examples of habitat management: eliminate or block access to food subsidies such as pet or bird food, keep garden waste picked up, etc., remove masses of vegetation or piles of material that allow nesting such as ivy). 
Ask your pest control company what type of bait is being used: Rats don't necessarily die outdoors, can be easy predation targets for both wild and domestic hunters (i.e., pets), can be a secondary poisoning risk to scavengers, and can end up dying inside/under structures. 
If you set lethal (snap) traps, be mindful of placement so that edge-traveling rats will run OVER the treadle, not INTO it, and place the traps so non-target animals (e.g., birds, cats, dogs, squirrels) can not get caught. Placing a "tent" of material over the trap, e.g., using parts of a box to make a roof over the trap whilst leaving the ends and travel path open, can help prevent unfortunate non-target kills and injuries. 
Finally, live-trapping and then release elsewhere is neither legal, nor ethical, nor humane. Rattus are non-native, invasive species on the American continents that degrade habitat for our native wildlife and affect human well-being wherever they occur. Similarly, domestic cats are not native species on our continent and are notably poor/ineffective at control of Rattus, but have severely negative impacts on multiple native wildlife taxa, so cats are similarly not an answer to rat management. 
I hope this information is helpful for you as well as your neighbors - Rat management is definitely a problem that requires cooperation across property lines!