Why does deterministic recognition of DYCK(2) languages in the streaming model take linear space?
Consider the restricted case in which the opening parentheses all come before the closing ones. Map ( and ) to 0, and [ and ] to 1. Then (after the map) an expression is well formed if and only if the first half is equal to the second half reversed.

Consider then a one-pass streaming algorithm that solves this restricted version of the problem in sublinear space. Using the construction in your question, to get a protocol for the equality problem with sublinear communication, Alice runs the streaming algorithm on her input (mapping 0 to ( and 1 to [), sends the working space to Bob, and Bob finishes off using the reverse of his input (mapping 0 to ) and 1 to ]). The streaming algorithm will output that the sequence of parentheses is well formed iff their original inputs are equal.