BYOD vs. CPD
If you're the one that buys the device, brings it into work, & use it to do your work, it's BYOD (Bring Your Own Device).
If your employer providese you with the device to perform your work -- whether it's a traditional desktop PC or a mobile PC (laptop, tablet, smartphone, whatever) -- then it's a CPD (Corporate-Provided Device).
For the latter, it belongs to the employer, so the existing data/Internet policies (& the associated software controls) already apply to it, plus they're the ones paying for the data plan.
For the former, there's the dual expectation that a) what you do in your free time (i.e. "off the clock") is up to you, but b) what you do during work time ("on the clock") is subject to monitoring & control by the employer. If that means your smartphone has a constantly-running app that logs when you access non-work-related apps and websites, & reports when you're doing it while on the clock... then that's fine, because your employer is paying you to work, not to do non-work-related items. And it's no more invasive than, say, tracking a delivery driver's truck to make sure he's working his route & not "taking an extra long break" at the mall; or, for example, requiring a sales rep in the field to call into the office every time he finishes one client meeting & is leaving for the next. You're on the clock, so your time is your employer's money. You'd be upset if a contractor was billing you by the hour to replace a damaged roof, & you found him taking a 30-minute "smoke break" every hour to pad his chargeable time; don't expect employers to be any more lenient when it comes to employees and their "work".
