Howdee! My first post here! I’m just writing along what comes to mind as I’m reading the two posts below.
First thought: privacy nightmare. Second thought: that might actually be useful. Third thought: if I can hack that, I can hear anything anyone says. Hmm, that must be my security mindset working overtime.
The difference with storing e-mail and I/M conversations, is that people generally think a bit more about what they write than what they say. You’ve got the chance to edit before you send written text. The equivalent is obviously thinking before you speak, which is a lot harder and less common. 
Searching through the material would require transcription software which could handle confusing input, because multiple people are speaking in a single conversation and the software might have a hard time distinguishing between them. Once the conversations are transcribed, you can do all kinds of exiting search operations on them.
Starting this as a phone application which records conversations would indeed be the logical first step. The question is though, is this not way too straightforward? Recording conversations isn’t really a revolutionary concept. If this is going to be distinctive, the added value will mainly be in the way the information is made available and accessible. Basically what matters is the question: how can you make this as useful as possible to people?
Still, the privacy implications are significant and so will the resistance against this be.
