Let's start by saying that we do not know what self-awareness actually is. We do not have a set algorithm for deciding whether species we encounter is self-aware or not. We are left with behavioral tests that provide circumstantial evidence, like recognizing oneself in the mirror. That said, any assesment of their self-awareness without direct quote might always prove to be incorrect. Let's try anyway.
In the begining, at the Battle of Naboo time, they were most probably not self-aware. As I mentioned in the comment to @Thaddeus' answer, being completely reliant on central command center to function removes that possibility. Command center might be self-aware, even intelligent, and imprint some of it's personality into the droids it commands - but that's pure speculation.
That apparently changed in subsequent movies, where no targetable command center is shown and yet all the droids seem to work. As Wookieepedia states:
Although the earlier generations were entirely dependent on Central Control Computers, post-Naboo models featured a great degree of independence and personality. Many older droids that were originally reliant on Control Computers developed personality quirks and a tendency to excessively comment on their situations.
However the reference they give for the first sentence is simply the third movie, which does not seem to be a strong evidence.
In my opinion, B1 battle droids never ceased to rely on command center - only on one in different form. How exactly is unknown (command network? redundant command centers on every capital ship with dynamic network topology?), but there is evidence for that in the Star Wars Roleplaying Game Saga Edition Core Rulebook
Battle Droids are drones controlled by a remote processor, taking their commands from a Central Control Computer (...) B1 Series Battle Droids can't be played as droid heroes
-Star Wars Roleplaying Game Saga Edition Core Rulebook page 199
That shows that being remotely controlled is an integral part of their being. They do not have the capacity of becoming independent "self-owned" droid - which is a prequesite for being a player character AND what looks like a good measure of whether they can be self-aware. That's in contrast with B2 series which
(...) relies on a central computer for guidance, but doesn't deactivate when it loses contact with the remote processor thanks to backup internal processor - a droid brain that takes over when the droid looses communication with the central computer (...) Super Battle Droids can be played as droid heroes
In conclusion: B2 series might be self-aware. B1 - most probably not.