NATS (UK) ATCOs, up until last year, had a 'Fam' (Familiarisation) Flight scheme whereby we could occupy the J/S and see what you guys did to earn all that money. (This included US based carriers for which FAA written authorisation was required and was, to my knowledge, always given).
It was highly beneficial to see what life was like at the other end of the headset and gave us an insight into CRM, aircraft performance, how to make life easier for busy crews, etc. The knowledge gained on such trips was invaluable as it enabled us to get a better understanding of what was reasonable to ask a crew to do to and what was/was not possible for them to achieve.
Our scheme has temporarily ceased due to budgetry constraints on the part of NATS. However, it would be a great pity if legislation on the part of the FAA/CAA were to prevent its recommencing at some point in the future.
(NATS currently includes at least two 'Fam-Flights' with a UK based carrier as part of its Student ATCO training programme)