Yes there should be private use of the internet. While there are some reasonable expectations of reduced privacy, it does not authorize the government to engage in large, aggregate data mining programs and data basing; nor does it mean that government gets carte blanche powers of search.
There's always things like facebook and such where people post alot of high visibility information. But even there it should be restricted. And certainly communications such as e-mail should be treated as an individual's papers and thus protected from unreasonable search and seizure.