Okay, this question has been coming up a lot recently in the comments section of this blog: What counts as previously published? And, in relation to this blog, does posting a poem in the comments of this blog mean it's "published"?
Before I begin, I think it would be beneficial for you to read this post from former co-blogger and Poet's Market editor Nancy Breen about the whole publishing question in "Published is Published!"
For Individual Poems
Many editors consider anything published anywhere at any time under any circumstances as published. This can even include public readings. And if a publication specifies what they consider published in their guidelines, it would behoove a poet (or any writer really) to respect the editor's considerations.
With such editors, a poem posted anywhere counts as publication, whether it's posted in a public forum or blog, or even a private, password-protected location online. In such cases, poems posted on this blog would be considered "previously published." However, there are editors who take a slightly different view.
Some editors consider a poem unpublished if it only displays on a personal blog and/or is in a "draft" form in a forum or blog. That is, if your poem on Poetic Asides is only a rough draft and not the final version, it would not be considered "previously published." If editors do not specify what they consider previously published, there's a good chance they fall into this camp.
For Poetry Collections
Except for rare cases, most editors/publishers of poetry collections accept previously published poems as long as the collection itself has not been previously published. Actually, the fact that poems are previously published usually helps in getting the collection published. That said, do NOT try to use poems posted on a personal blog or public forum as a publishing credit. Such credits hold little weight, since there is usually no screening process, because eveyone can get published.
My main point here is that individual poems that are considered published by journals can still be considered unpublished as components of a poetry collection. And that even individual poems that are considered published are welcome in "original" collections of poems.
In fact, "new collections" can be made from selecting poems from previous full-length collections and chapbooks.
So, How Should Poets Proceed?
Armed with your knowledge of what is and is not considered published, you've just got to pick your battles and act accordingly. For instance, most of my poems are not published on my blog, because I want to have as many publishing options available to me as possible. I share drafts of these "unpublished" poems with close poet friends to solicit feedback for revisions.
The poems I post as parts of prompts, I consider "published," though I would not use it as a publishing credit if I tried including any of them in a collection, because I also consider my poems on this blog to be "vanity publication credits." I make an informed decision to write a poem a week just for the act of creation.
Considering how much money most published poets make anyway, I don't view this as such a bad decision. But every poet has to make this decision on their own.