Verizon admits that the information will be shared within "the Verizon family of companies," including that of (Engadget parent) Aol. From there, the data will be used to "provide more relevant advertising within the AppFlash experiences and in other places." The other places being a euphemism for banner and display advertising all across the web.
So, if you're trying for a baby and you've got a fertility app on your phone, it's reasonable to expect plenty of banner ads for diapers and formula feeding. If you're doing something more private, like making your first steps out of the closet or dealing with a substance abuse issue -- and you've got a relevant app -- then Verizon's gonna know about it.
To be fair, Verizon justifies its stance by saying that it'll need some of this data in order to make on-demand services work. How, after all, can it seamlessly tell you local movie times and call you an Uber to the cinema if it doesn't know where you are? Not to mention that Google already snatches most of this information for its own purposes.
But, as the EFF points out, most of the Android devices on Verizon's network will now have a common app that hackers will be probing for holes. Should a nefarious type find such a vulnerability, then you can be sure that same personal data will be sold off to the highest bidder.
Update 2: Following Verizon's statement, the EFF has actually taken the step of withdrawing its prior accusation of the cellular network's motives. The privacy body has pledged to investigate the matter further, but it looks as if it may have been a lot of fuss over what amounts to very little.
Update 3: Verizon has also posted a brief explanation on privacy in light of Congress deciding to roll back the FCC's privacy laws this week. "We have two programs that use web browsing data -- and neither of these programs involves selling customers' personal web browsing history," chief privacy officer Karen Zacharia said. "The Verizon Selects advertising program makes marketing to customers more personalized and useful -- using de-identified information to determine which customers fit into groups that advertisers are trying to reach."