After some experimentation, I've noticed the following:
Monsters do indeed level up with you. You can see what level the monsters are currently by looking at the top right of the screen:
The easiest way to tell monsters are gaining in power is by turning your difficulty up to something like Torment and looking at their max health (assuming you have health bars and numbers enabled).
Any monsters you were fighting when you leveled do not appear to level. I've escaped to town, hung out in town for a long while, and popped back to see the monsters I've fled from at the exact same HP/Max HP I left them. However, when I started engaging new monsters that were off the screen, I observed the new creatures had much more health than previous mobs of their type. This behavior is different from when another player joins, where after several seconds all monsters gain a proportionally larger amount of health.
Monster health starts scaling up pretty fast, especially once you reach the 50s (and presumably in the mid-to-upper 60s). Thus, the difficulty gets harder as you are quickly gaining levels faster than you can acquire all-new gear and the bad guys are becoming harder to kill. The massive health/damage bonus of Master and Torment difficulties amplifies this effect.
Monsters are the same level as the game's creator and level up with them. Monsters never de-level. For example, if a 23 Monk joins a game started by a 30 Barbarian, the monsters will be level 30, even if the Barbarian leaves. They will remain level 30 until the Monk hits 31, and then will start gaining levels along with the Monk.