Does the presence of Diagnostic Data mean someone unauthorized is checking my phone? What does Settings > Privacy > Diagnostic &amp; Usage Data mean? 

I think there is a chance that I am being watched.
Nobody is watching your phone (at least from this specific point).
The setting you are talking about is called Diagnostics &amp; Usage. It looks like this:

It explicitly says what it does:

Help Apple improve its products and services by automatically sending daily diagnostic and usage data. Diagnostic data may include location information.

If you feel like you do not want to share your iPhone application usage data then just check Don't Send.
Note that diagnostic data will be saved, but not sent to Apple, on your device in any case even if you choose Don't Send.
If you proceed into Diagnostics &amp; Usage Data then you can see what application and when saved its data on your phone. What looks something like this:

In case you want to find more information about this topic, consider visiting this official Apple page.