RedKali: I took too long to post this, as you seemed to have moved on, but I hope you'll consider it anyway, because it might help with offering explanations or possibilities to curious students:
No, I don't think that "If two people observe the same series of events, in the correct sequence, and accurately describe the elements involved in the dream, yet neither notice one another," that ought to be considered a shared dream. There are other things, perhaps even more likely things, that can make dreams similar, or cause them to be remembered as similar.
Originally Posted by RedKali
One thing I've often noticed in (usually breathless) accounts of shared dreams like this is that the two dream-sharers are almost invariably involved in very similar waking-lives. They may be family members, close friends, or in intimate partnerships, have closely shared dream goals, or something else that causes them to have very similar life experiences and attitudes. So there is an excellent chance that they are generating very similar waking-life experience, like day residue, expectation, possibly even thought patterns or perceptual preferences/biases as well. This proximity of experience could lead to unshare dreams that seem the same; even, perhaps, to the degree you describe above, RedKali. Though this similarity is certainly a sort of sharing in itself, I'm not sure if it's the sort of dream-sharing we're interested in teaching here.
The same goes for planned or anticipated shared-dreams. Two people hell-bent on having a shared dream with each other might very likely incubate the same sort of dream, so that when they report to each other the next day it isn't hard to find things that match...even if nothing does: I can't tell you how many times I've heard or read reports of shared dreams where each dream has nothing to do with the other, but the two excited dreamers find or invent all sorts of things that in their minds surely prove they were sharing a dream.
I think that, since dreams are essentially projections (literally projections, in dream-sharing, I suppose) created by an individual person's mind, the presence of that mind ought to be noticeable when it encounters the presence of your own mind during a dream. In other words, I think that what you are really sharing in a shared-dream is a sort of meeting of minds (or individual presences, perhaps), with the dream scene itself being secondary. In a sense, the presence of another mind in a dream would precede matching imagery, with that similar or recognizable imagery forming only after one dreamer begins paying attention to another dreamer's presence.
For instance, a shared-dream might start with the appearance of, say, a curious point of light on the horizon on your dream: you sense something different in it, something that is not you, so you focus on it. Your attention on that light brings your mind into closer sync with it, and eventually that point will expand into something more elaborate, perhaps even a new dream scene, that assures you that you are not alone in the dream, and allows you to communicate through imagery or words with the other presence. Which brings me to another point:
You wouldn't need to see an identifiable dream body of your dream-sharing partner to know she is present, either. Indeed, given that we really aren't built as television transmitters, if dream-sharing occurs at all it very likely does not include an ability to tell your dream-sharing partner through transmitted imagery what exactly you look like, or even what you wish to look like. Instead, your presence will appear as a construct created by your partner's own dreaming mind, and that construct will very likely, at least initially, not be a facsimile of you -- or anybody; a "receiver's" dreaming mind might initially produce you as, say, a fire hydrant, or perhaps an angry sloth... the construct created could be anything, which is all the more reason that acknowledged presence is more important than shared imagery. Later, after communication has begun, you and your partner's avatars might start forming into something more recognizable, though even then you will likely appear much different in the dream than you do in waking life.
So, and as a tl;dr:, I guess for me presence precedes similar imagery in a shared dream, and, while shared presence (when noticed) cannot be mistaken for a shared dream, imagery can, no matter how similar it may be.
Also, and almost as an aside:
I think lucidity is not only helpful, but may be may be an imperative for successful shared-dreaming, and I do so for a singular, if odd reason: The mechanics of shared-dream are impossible, to me, unless we are all sharing our dreams all the time.
Speaking of that, I don't think people need to be lucid to have a shared dream. The elements (which are still being discussed) of a shared dream can be met without having to be lucid. Though! Lucidity certainly can be helpful.
It makes much more sense to me that our dreaming consciousness exists, space&time-free, as an individual thread in a great fabric of say, collective consciousness or perhaps thought energy, where every thread is in constant contact with every other thread, and when dream-sharing we are not finding our partners as much as we are singling out their individual presence --their thread -- among the weave of dreaming thought, than it does we that are somehow magically able to find someone (even across time-zones of physical world) among the minds of billions of sleeping people.
If all our dreams are somehow connected in an as-yet undiscovered metaphysical matrix, it would be safe to say that, by any definition, we're all dream-sharing every time we dream. We just don't know that it is happening, either because we are hard-wired to ignore it as a sort of anti-confusion defense mechanism, or due possibly to the psychic strength our own dreams tending to overwhelm the influence of other people's passing dreams; or maybe we just don't think of it at the time, just like we don't think of so many other things when not lucid.
The presence of lucidity, though, allows an opportunity to pay attention to our dreamworld, to consciously (if metaphorically) examine that weave of dreamers to spot the dreams of our dream-sharing partner, and also to acknowledge when our partner is looking for us.
So, in effect, if dream-sharing can happen at all it is just as likely that we are all sharing our dreams all the time, but we don't notice it during the dream, which sort of makes such sharing irrelevant from a consciousness perspective. With lucidity, though, we can learn to pay attention to the cosmic conversation and both seek out desired individuals or make ourselves receptive to contact by individuals.
This sounded a lot better in my head, and I know it is meaningless to the discussion, if not altogether meaningless, but I'll leave it up anyway... maybe one day it will matter.