It's been awhile since I read the various publications about the L&HR, so my memory may be "off". Not to directly contradict C_F, but I understand...
The PRR (and later PC) actually had trackage rights to run there own trains/crews Belvidere to Maybrook. This came about when the L&H gained their trackage rights, from the PRR, down the Bel-Del from Belvidere to Philipsburg to reach the Delaware River Bridge and the CNJ/LV at Easton. The era in which these PRR run-throughs began isn't clear, but 1963-65 may have been possible. I recall late PRR ('68) symbol freights NE-2 and NE-1 ran the L&H to Maybrook, as well as PC era run-throughs. I also recall the PC era run-throughs were due to the poor condition and congestion of the ex-NYC River Line. I have a PC system map showing the L&HR route Belvidere to Maybrook as "Overhead Trackage Rights".
Somewhere in this website, (PRR or PC Forum?) a former PRR(PC) employee, "TSP-5 TSP-6", shed some light on this topic.
The CNJ "rights" were not "trackage rights", but were the result of the agreement reached between the L&HR and CNJ concerning employee's work. The CNJ couldn't just run their own trains/crews over the L&HR whenever they wanted to. Because the L&HR ran to/from Allentown over CNJ track between Allentown and Easton, the CNJ was "compensated" by having CNJ locomotives and crews run over the L&HR to Maybrook. When the L&HR mileage over CNJ track added up to a run over the L&HR mileage to Maybrook, the CNJ ran their own locomotives and crews over the L&HR to "equalize" the mileage of each road's portion of the run. Keep in mind, the locomotives and crews weren't "paired up"; L&HR crews used CNJ units and CNJ crews used L&HR locomotives. Pictures abound of mixed L&H/CNJ locomotive consists, so the locomotives weren't an indication which crew was running the train that day. This practice ended when the L&HR quit running to Allentown and interchanged with the CNJ at Hudson Yard, Phillipsburg NJ, circa 1971. No L&HR crews on CNJ rails = No CNJ crews on L&HR rails.
Making an educated guess, the formula most likely worked:
Allentown to Maybrook = 111.2 miles; 16 miles was CNJ.
7 runs by L&HR crews/locomotives over CNJ rails meant the 8th run would go to CNJ crews/locomotives. So, every 8th day, AO-4 would have a CNJ crew out of Allentown and they'd return out of Maybrook that evening on OA-1.
In addition, RDG units were used on the L&HR. While the Reading officially ended at Allentown, it's most likely the RDG units made their way onto L&H rails as a result of the CNJ "borrowing" them and using the RDG units on the L&H Maybrook jobs, as well as run-through unit coal trains. I haven't seen any photos of mixed L&HR/RDG locomotive consists, but have seen CNJ/RDG mixed as well as solid RDG consists, which leads me to believe the RDG units were "borrowed" and run-through.
Keep in mind, there's a significant difference between the PRR(PC) and CNJ agreements with the L&HR:
The PRR(PC) had the right to run whatever/whenever/whomever they wanted over the L&HR Belvidere to Maybrook.
The CNJ was "restricted" to running crews/locomotives Allentown to Maybrook based on the mileage formula...
I've never read of a similar agreement between the L&HR and DL&W addressing the Port Morris to Maybrook work. As far as I know, the L&HR had trackage rights over the Sussex Branch Andover to Port Morris without any reciprocal DL&W(EL) rights over the L&HR. This "trackage rights" agreeement was limited to strictly through trains that originated out of Maybrook (NH) destined for the DL&W and vice-versa. Any cars that originated on the L&HR for the DL&W (and vice-versa) were restricted to be interchanged at Andover Junction Yard.
In summary, the PRR(PC) had trackage rights over the entire L&H unrestricted. The CNJ had limited rights over the entire L&H to supply locos and crews based on a formula. No other railroads had any rights/agreements to "set foot" on L&H property outside of yards used for interchange, such as Hudson, Andover, and Greycourt.
Please add/correct as required, since my knowledge of these operations is second-third-and fourth hand information...