This is an excellent question.
Authors of novels regularly use full stops to end non sentences because the effect of a full stop (period if you're American) is required to achieve the desired flow for the reader. However, there is no grammatical basis behind this. Both of your examples are difficult to justify grammatically. However, I judge the second one (In regard to what, the communication between us?) to be the least correct. This is an example of a run-on sentence.
The second half of the first example (In regard to what? the communication between us?) is not a sentence, but - as explained - there is some leniency in this regard, especially with interrogative sentences (i.e. questions).
However, I would insist on a capital T on 'the' after the question mark.
