I wouldn't mind making an endpoint like /member/{id}/eligibleForFreeTraining and just return a true or false but that is not CRUD or REST.
There's precedent for that; for instance:
https://developer.github.com/v3/gists/#check-if-a-gist-is-starred
I could possibly add this information onto the Read endpoint but that means that its going to get returned on every read call. This is fine this time, but what happens when there are 100 checks and now we return 100 fields that are only needed at certain times? It just gets bloated.
One answer is that you can have multiple representations, that have different levels of details. The RESTful way to do that would be to have different media types that describe the different representations.
For example, the Apis for the Sun Cloud defines a number of different JSON representations, identified by distinct vendor media types.
It's always been allowed to implement resources with more than one media type representation, and nothing in the "rules" prevents you from having more than one representation with the same media type suffix.
The client request for the resource would specify the appropriate media type, and the server would offer it up, if available.
The RESTful way to communicate about the available media types is to announce their availability via hypermedia controls. In other words, in addition to providing the client with a link that specifies the "standard" media type for the resource, you provide it with alternative links that it can use in other circumstances.
EDIT allowed, but not necessarily encouraged.
We encourage resource owners to only use true content negotiation (without redirects) when the only difference between formats is mechanical in nature.
-- Fielding, 2006
That would favor, as Jack noted in the comments, using a different logical resource for the different representations you want to support.
That said, it's probably worth pointing out that in a REST application (which is to say, if your API is providing a protocol), you wouldn't need to tell the client about the state of the flag, but instead use the state of the flag to determine whether or not to include links to other resources in the representation of the resource that you send to the client.
For your example, that would mean that you would include an "Enroll in Free Training" link to eligible members, and withhold it when members are not eligible, with the understanding that the client only acts on those links that have been provided to the client as part of the application state.