clearance is needed
First you need to be specific about where your film is going. Are you trying to sell this film? Are you doing the festival circuit? Or are you just showing it privately? Is your film narrative? Or is it documentary?
Let's start with the most strict scenario: if you're planning on selling your narrative film or allowing your film to generate a profit for someone, then you need to obtain written clearance for all the logos that appear in your film. You also need to clear people's faces, too -- as in have them sign photo releases -- but hopefully that's obvious and you already did get the talent releases. For logos appearing in your film, the buyer (ie the distributor) is going to demand that you provide documented product clearance. This is part of the "deliverables." That's just a fancy word for what you'll give the distributor: your completed film, the alternate formats for visuals and audio, and all the paperwork.
In summary, you do need clearance if you're selling your film. If you don't get clearance, you will need to blur out all the logos. Depending on how you shot the film and how your post-production work flow is set up, this may not be too difficult to do. You can use motion tracking and so forth.
Would those big companies ever go after you? They might.... just to make an example out of you. In general, big companies don't litigate unless there is big money involved. So if your film is a useless flop, many companies won't care, especially if you're not showing the product in a "bad light" in the first place. However, if your film gets popular, you could run in to a nasty predicament down the road.
To get PAID by these big companies for product placements, you'd need to first make sure your film depicts their brands in a favorable light. (This is a figurative term, not a cinematic term.)
You can still get clearance after the fact, but it's not a guarantee at this point.
And let me just state for the record that none of what I'm saying is legal advice. You have to consult a qualified attorney on matters like this.
Shanked