The muted do not know (but may suspect) that another member has muted them.
But they can't be certain unless that member tells them. Either in a childish get-in-the-last-word of "muted" or something similar.
The mute is just a single click way to enforce a "you know what, I am never replying to this person again" desire. People always want to "get the last word in" or "win the argument", as if such a thing is/was possible.
So no, if you've muted someone, or someone has muted you, the only way that the muted person will know is if they are told by the person who clicked on the button.
You should see (unless you are running scripts) collapsed topics in the main listing (those will be topics where a muted member is the OP). You can choose to expand them and see if your reason(s) for muting still exist.
Also, when in a topic, if there are posts from a muted member, there will be a link at the top of the OP allowing you to "show hidden posts" and those who are on your mute list will be shown on that page.
I personally review my mute list (it's in your profile settings) and check the recent posts of members that I have muted. If their recent post history reminds me why I muted them (showing that they've not altered their posting habits, which is fine) then they stay on my list. If I can't see anything in their recent posts that reminds me of why they got muted, I will usually unmute them (people can change, and if they do, I am happy to see their posts again).