Is there a piece of teleconferncing equipment that records and
transcribes meetings? I would like a device that can listen to a
conference in an office boardroom, digitze the conversation and then
produce a text document of what was said, preferably by whom.

An acquaintance of mine tried to do this using a well-regarded speech
recognition system called Dragon Naturally Speaking. The results were
not very satisfactory, since the transcript required an immense amount
of editing for errors that a human transcriptionist would never have
made. For instance, the system heard the term "IT," meaning
"Information Technology," and transcribed it as "eyeteeth."

A big problem is that business meetings typically have many overlaps
of conversations. When more than one person is speaking
simultaneously, this confuses the speech recognition software.

Another problem is that this kind of software requires an extensive
"training" period for each user. The "training" is not to teach the
humans, but to teach the software to recognize the unique
characteristics of each person's pronununcation. Voice recognition is
much less accurate if there are attendees with whom the system is
unfamiliar. It will work best in business meetings that always involve
the same individuals, with no newcomers.

There's a useful article here. The references to hardware are out of
date, but much of the info about speech recognition still applies:

http://project.uet.itgo.com/basics.htm

But what if people could introduce themselves and say a few indicative
words to identify themselves initially... I am not so interested in
the transcription as much as recording the data and having the system
identify who said what and maybe even making an annotated recording.
Basically leveraging what computing devices can do well, now.

Most areas will have shorthand writers or stenographers who specialise
in transcribing meetings.

Owain