On February the 15th 2003, millions of people marched against the Iraq war across around 800 cities - with possibly as many as 1 million in London alone. This made it into the Guinness Book of Records as the largest ever anti-war rally.
What happened? We went to war.
Demonstrations are not some guaranteed method that will always bend those in power: it's just one of the few methods that affect change and have in the past.
If demonstrations were all that it took every time, then the French wouldn't have to behead their king, and the Americans wouldn't have to fight a revolution.
It's also about how seriously those people mean it. Marching for a day is useless if you don't also change how you act politically, inform people, change your buying habits (e.g. boycott some company), vote differently, go on strike, etc.
Else, who cares if they protested? They just had their fun for a day and went back to watching their favorite TV series. They didn't even mean it that much themselves.
Protesting is not just standing around with some banners. Demonstrations, strikes, occupations, etc. are all part of it, and have brought down many a government.
It's not about "conscience" -- except if we mean some tame gatherings where people stand quietly and hold some picket signs for a few hours and then go home.
You'd be surprised. Maybe read some history. Except if you mean: it doesn't work to guarantee an utopia or even that the things/rights/etc won by protesters are secured for eternity. Then again, nothing does.