Regulation 11 transposes Article 11 of Directive 2014/24 almost verbatim, word for word. The Regulation excludes from application of Part 2 any service contract that is awarded under the basis of an exclusive right held by another contracting authority. Said exclusive right needs to be conferred by law, regulation or administrative provision compatible with the TFEU.
There are two important components to take into account with this provision. The first is that this exclusion covers service contracts between two different contracting authorities, so any organisation or institution considered to be a contracting authority could benefit from this exclusion. As such, the way I see it, a contract involving as counterparty a contracting authority as defined by Directive 2014/25/EU would also generate this effect. It could be questioned however, if said entity was not operating in that specific contract as a "contracting authority" if that exclusion would be applicable or not. Perhaps not as exclusions to a general rule (in this case, coverage or scope) should be interpreted restrictively.
The second issue to take into account is that the exclusive right awarded needs to be conferred by law, regulation or administrative provision and be compliant with the TFEU. I am not entirely sure if this specific reference to the TFEU was needed as this is probably another tautologic reference to the Treaty. In other words, its absence would not change the need for compliance with the TFEU. In consequence, any exclusive right which violates the TFEU (say for violation of EU principles) will not generate this exclusion. As usual, the problem will always be transparency and accountability as it is unlikely interested parties will be aware of i) the right; ii) that a contract exists; iii) the right violates the TFEU in any count. Therefore, the question I would like to leave hanging is: how can the contracting authority comply with requirements such as equal treatment and non-discrimination or transparency when awarding these exclusive rights? Do they need to follow a due procedure of sorts or can they simply "award" the right to another contracting authority? This can have an impact on how contracting authorities organise their collaborations in procurement for example (see next paragraph).
Albert highlighted in his post a small difference in both texts. Whereas the Directive allows for the right to be held by an "association of contracting authorities", Regulation 11 includes no mention of them, leading Albert to conclude that there is a restriction in place. I am not so sure as in England and Wales an "association of contracting authorities" can be constituted as a new centralised purchasing body or on a shared services/category management type of model (where one authority leads), both of them constituting a single contracting authority for the purposes of the Directive, so there is no need to provide for an "association of contracting authorities". As a speculation, that would be my explanation for the difference.
In addition, Albert raises a good point about the functional irrelevance of this exclusion and the conflict between Article 11/Regulation 11 and the principle of competition. The exception afforded by this Regulation contradicts said principle of competition by extracting from general public procurement rules specific contracts. Having said that, however, the specific reference to the compliance with the TFEU ensures that any violation must be kept to a minimum.