Yes, absolutely they can be members of both.
ICBI, in one form or another has been working away in the background. It's only in the last year or so that we are actively seeking new members (now that we have a structured organisation), and most new members would already be members of Beer Ireland. It is not a them or us scenario by any means.
To clarify the ICBI logo (there will be a launch and press release soon to explain it better) to use the logo you do not have to be an ICBI member, just fit the criteria.
Provenance was a big part of the discussion when setting out the criteria, so on that side the beer that carries the logo must be brewed in your own brewery (ie not contracted) and packaged on the island of Ireland.