I'm quite sure they do and this is probably a moot issue as we have 8000 employees on company phones. They wouldn't have the manpower to check behind all of us but it concerned me enough to ask.
Much of what Ed Snowden has been claiming is still a bit vague, but it's certainly clear that many of the more feverish conspiracy theories are more credible than most us thought possible.
But you don't work for the NSA; your employer hardly has NSA-class capabilities. You're probably safe; iMessages are strongly encrypted, end-to-end, and it's highly unlikely your employer can see them without physical access to your phone and knowledge of its password (You DO have a password-lock enabled, don't you?), OR your iCloud username and password-- which is really all that's needed since any Mac or iDevice can be linked to your iCloud services (including archives) with that info... guard it carefully and mind any alerts from Apple regarding activity on your account.
But as to your point that you are safe in the herd, a word of warning: While you are undoubtedly correct that no one is watching you on an hour to hour basis, some companies do do spot-checks. On computers, it's distressingly common nowadays for companies to install screen-snap, keylogger, geolocation-logging and webcam-snapshot utilities to record employee behavior... not so much to enable 'round-the-clock surveillance as to enable occasional validation of employee (and consultant) activity, and to build a repository that might be useful for disciplinary, termination, legal defense or negotiation-strategy matters should an employee merit attention for other reasons. In other words, your repository of recorded activities and behaviors might not be looked at until you're in trouble for other reasons or have the company in a tight spot in some way or another ...and then you'd better hope your brother's nudie pics aren't available to them, that you've worked diligently at your computer and don't make a habit of surfing non-work-related sites or playing games on company time, that you didn't go skiing when you said you had the flu last winter, etc.
Bottom line, don't trust a machine or a service that you don't control. You have ceded partial control of your iPhone, so that's a concern. You have NO control over machines you did not purchase and configure yourself, and you should have no expectation of privacy on the company network (or any other) unless you favor services with good crypto hygiene-- like iMessages. All bets are off if you lose physical access to your devices, and who knows how deep the tentacles of governments (not just America's) really can go even if you do everything right.
I speak as a guy whose computer was physically hacked in a very interesting way when I left it in my hotel room in a country well-known for its ...inquisitiveness, shall we say. I've become something of an encryption evangelist as a result. It's a rough world out there, and anything you say or do can be used against you. Ideally you'd manage your own encryption rather than relying on any third party to do so for you, but as far as those go, Apple's encryption is better than most.
This is a very important and sobering topic, and I thank you for bringing it up.
