One of the most dangerous aspects of Venom is that he doesn't set off Spider-Man's spider-sense. The explanation for this was always that it had once been his costume and being in proximity to him, his warnings did not register the symbiote as a threat.
But when Spidey first gets the symbiote, it does set off his spider-sense as seen here:
What the heck is that--? And why is it making my spider-sense tingle?
So what happened?
Marvel.com sort of handwaved the explanation in a previous version of their bio for Brock:
Brock taunted Peter Parker with minor assaults, such as pushing him in front of a moving subway train. The alien costume could cancel out the wall-crawler’s spider-sense presumably by projecting conflicting frequencies upon Spider-Man's brain waves.
The generally accepted answer you'll find discussed is that the symbiote takes on DNA or abilities of its hosts. Peter's Spider-Sense does not register non-threats, and the Symbiote exploits this by somehow (see the "presumably" even in the Marvel.com description) confusing Peter's senses by appearing to be a non-threat or even Peter himself.
Spider-sense doesn't affect clones of Peter Parker or people he trusts (not exactly scientific). Here's some semblance to an explanation but I think it's because it adds suspense to the story.
