Google returns results for both variations of this common phrase.
What is the meaning of this phrase, and which one is correct? Is the speaker carrying their shame or are they exposing it? Or is this just a case of two separate meanings?
First of all, it helps to enclose the phrase in quotes when googling:
Secondly, having looked through the first 10 pages of the Google results for "bare the shame", exposing would not work in most of those contexts at all, it's quite obviously carrying. And quite a few of those contexts feature extremely poor grammar and punctuation.
So, to me, the answer is pretty clear: "bare" is simply a common misspelling of "bear", whether in the context of this expression or elsewhere.
The common phrase is 'bear the shame', as in carrying the shame. The prevalance of 'bare' on Google is probably a result of misspelling. You would probably say something more like:
He laid bare the shame of corruption.
Something interesting to note. I don't have any verification of this, but I'm pretty sure the original root for this phrase is from the Bible. When Cain killed Abel and G-d chastised him, he (Cain) says "גָּדוֹל עֲוֹנִי, מִנְּשֹׂא" (gadol avoni minso) - My sin is too great to bear. The word מִנְּשֹׂא (minso) literally means to carry.