It's not the boom, boom 90's so it's doubtful you'll see a hostile takeover. If they were to attempt it a BP would start talking to the largest shareholders to see if they were amenable to it and see what price point they'd need. If they had enough interest they would publicly announce they wish to buy the company and at what price. Then it would be up to the shareholders to vote on it.

Sometimes you'll see private equity firms or hedge funds try to accumulate enough shares and then try to force a quick sale to the highest bidder for a quick profit. That none have tried to pick up a lot of cheap shares that's not happening now. Once market cap hits $10 billion then risk vs. reward may become too great for that.

For a normal buyout you won't know it's happening unless there's a leak or the company announces it and sets up a shareholder vote.