Yes they can, but it's a little more complicated then that.
Firstly, the main reason a "web-spinner" doesn't get caught in his own web is because spiders use tiny curved claws to make contact with the web itself.
The more familiar they are with the type/layout of another spider's web - they less likely they are to get stuck.
However, if you were to pick up a spider and throw it back into its own web, there's a good chance it'd get trapped.
Secondly, not all spiders have sticky webs. An example would be the grass spider - but that doesn't mean insects can't get their legs tangled in its thick strands (including other spiders).
Spider on spider murder rates are also an interesting topic.
Certain spiders (like the friendly Orb spider) actually share webs. Other spiders have no problem attacking each other.
Here's a fascinating fact though:
Black widows, known for their venomous bites and sexual cannibalism, actually have an enemy in the common Daddy Long Leg.
Daddy Long Legs are known to rush straight into a Black Widow's web unimpeded and overpower them with ease.
Here's a video that will make you think about Daddy Long Legs in a whole new light:
