The thing is, yes you can be held responsible. So it's a good idea to not allow copyright material and remove it when you find. Otherwise it's possible in some niches for your users to think there is no problem in doing it and before you know it the site is full of infringements and file sharing.
So it's kind of self proliferating. If your site is making money, I imagine that worse things than just DMCA takedown might happen.
OTOH, you don't want to stifle user activity. I tend to turn a blind eye to small things, but if I notice something with an obvious copyright (e.g. Getty Images watermark) then I remove it.
I tend to think Youtube embeds are OK, as Youtube should be policing those.
Thanks. Yes, I usually discourage people from posting any copyrighted image and any text from other sources. If they had to copy something and is not sure if they have CC (or similar) license, I ask them to at least cite the source.
But even then, it's not always possible to filter out those (especially some tends to quote texts from other websites, as it is), so am a bit concerned if someday someone sends me a legal notice without any takedown notice or warning.