I saw a sign at a recent protest that read "Silence is Consent". Prior to this, I've been blasted all my life with "Silence is Not Consent".

So which is it? Can you help me out here?

As for myself, I'm certain that silence is not consent. It's a freely-made choice not to take part in something.

Thus, doing nothing is not equivalent to picking winners. Picking winners is when you ... wait for it ... pick winners.

In my above example, by doing nothing you pick winners.

Not in all situations, doing nothing is doing nothing. Take the famous trolley problem. By doing nothing you pick 5 people to die, if you do something you pick 1 person to die. One of those will come true, either inaction or action will determine which. But that also means your inaction is an action of yours to force an outcome. This doesn't necessarily mean it's the outcome you want or the outcome that is preferred. Inaction can be the worst type of action to take in some scenarios, but it is the action of non-intervention that creates an outcome regardless. Inaction doesn't mean you consent to the outcome either.

Sometimes, doing nothing is even illegal, eg not helping someone who is in danger when one can do so at no risk, or sometimes, not reporting an ongoing crime (I think they can be illegal, here where I am)

Both forms of “Silence is (not) consent” are true, they just refer to different things. If you are in a position of authority, silence is effectively consenting to the status quo remaining unaltered.

If you are not in a position of authority, whoever is in that position can’t take your silence as consent of their choices.

Nice explanation; I never thought about this in such a logical way (although intuitively I'd known the answer)

I think in your example, whichever use of silence is addressed to a specific audience.

For the former, the idea (I think) is that if you don't consent to something, speak up to make a difference.

In the latter, it's more that one should not assume consent on the part of another, presumably in the context of a sexual partner.

It can be both, depending on the situation.