The watch battery inside is called a CMOS battery and is most likely something you will want to keep in if you use your device daily. It keeps the time in your device even if the device loses power, including the time change (meaning if you unplug a device for 2 hours, the time will still be correct, not the time when it lost power).
Also, many (especially cheaper) devices use CMOS batteries to keep system settings alive. This may not be true of Android (probably all stored in internal storage), but for other devices this is true.
For instance, my PS3 got the YLOD, and I learned how to fix it. To do so, I had to take it COMPLETELY apart to get to the CPUs, and to do that, I had to unplug the CMOS battery. When I would start it up, it would give me a screen similar to when I first got the PS3, asking me to set things up.
Like the other user said, it isn't enough power to spy on you when the battery is removed. Besides, that would require a LOT of control over your device. All to look up at the ceiling and hear hissing noises. If you want to make sure people aren't spying on you, don't install any shady-looking applications (they could be malicious). Use applications from trusted sources like Google Play, Amazon, Slideme, and if you're not sure about something, look at the reviews and ratings. although chances are, nobody will spy on you unless it's your phone's manufacturer/carrier (and they add it into the firmware) or the Gov't with some secret wiretap laws/court orders. With the firmware thing, it'd probably be safer to use a custom ROM than an OTA update... those random reboots I used to have were fishy......
