Our by-laws name certain committees as committees of the corporation --- "No such committee shall have the authority to bind the board. Members of such committees of the corporation, who may be non-trustees, unless otherwise designated, shall be appointed by the President."
Can the non-trustee members of a committee vote if one is called for in the committee? It seems like they could because the committee can't bind the board, however we could see where their vote within a committee might mean that something isn't brought to the full board.
Related to this, should we amend the by-laws to specify them as voting/non-voting members of committees?
This reply will answer the questions up-front, and then tackle the concern about the full board not seeing a matter since it was voted on in committee in the "background and commentary" section.
1. Can the non-trustee members of a committee vote if one is called for in the committee?
Yes, if a library board has a committee with non-trustee members, those non-trustee members can vote.
2. Should we amend the by-laws to specify them as voting/non-voting members of committees?
No, there is no need to amend the bylaws. If the non-trustees are properly appointed committee members, they may vote. If the non-trustees are not actual committee members but are instead there in an advisory capacity, they should not be referred to as "members" in the first place, and the appointment letter should make that clear.
And now, for some background and commentary.
Back in 2014, the New York Not-for-Profit Corporation law was amended to create two types of board committee:
As the member points out, "Committees of the Corporation" are committees that--while they might work hard on matters of great importance to the library--cannot "bind" the board...meaning, they can't make "official" decisions final (authorizing a budget, signing a contract, or voting to hire a director).
"Committees of the Board," on the other hand, are authorized to "bind" the board in certain matters, including investments, endowments, employment, and some matters relating to real property.
While this "Committees of the ______" change was quietly revolutionary in many not-for-profit circles, it was already somewhat familiar territory for libraries, because the Education Law already authorized them to have an "Executive Committee" to "transact business of the corporation" between meetings. In other words, libraries were already used to designating committees with express and binding authority. This just gave them more options to bring on more participants who were not trustees.
Now, while "Committees of the Corporation" may not be not tasked with the Really Big Decisions, as the member points out, they can still do very important work.
For example: let's say a library has created a "Public Relations Committee" ("of the Corporation") responsible for monitoring and identifying tactics for the library's presence in traditional and social media. The committee doesn't sign contracts or even write press releases; it simply monitors and issues advice, meeting virtually once a month to review the library's media footprint and track its various metrics.
Based on some observations, the committee decides the library should adopt a "Social Media Policy." Since the committee can't "bind the board," they can't vote to adopt the policy, but if they vote to do so, they can:
Now, here's where the member's concern comes in: How does the full board know this work is being done?
The critical work of committees "of the corporation"--even if they are not "binding" the board--should be connected to that of the full board by a routine report (or meeting minutes) that are "received and filed" by the full board subsequent to every committee meeting. That way, whether or not the committee votes, the board is aware of its work, and what is in the pipeline. Committee work should never take place in a vacuum; it should always be linked to the operations of the board by reports and minutes.
A high-functioning library board operates like an orchestra--different sections may rehearse separately, and sometimes, there may even be a prima donna moment or two--but the end goal should be harmony.
A board's various committee types may have different functions, compositions, and authority, but they are all part of that system. Thank you for a great question that showcases their differences and value.
 A valid concern.
 They can't do everything, though, so proceed with care!
 A wild bunch who brandish the word "fiduciary" the way some people wield the term "linebacker."
 The special focus of a PR committee, by the way, is why a library may want to bring on some non-trustee "ringers" to help with specific issues (a building committee is another great committee where you might want someone for a non-specific set of skills).
 Or at least, whenever the committee meets. Some committees only meet and act once a year; obviously, that committee only needs to submit one report!