To put it simply, no. A Twitter user has no means of knowing who is watching their account or individual tweets; there is no Twitter search for that. Direct engagement—a comment, a favorite, or a retweet—is the only way to know for sure if someone has seen your Twitter page or postings. However, through data mining and some simple math, all sorts of tools are available for anyone to estimate how many people are reading their feed.
The most common method involves counting the number of unique IP addresses that are viewing a user's timeline page. This is called "viewing count" or "click count." The problem with this approach is that every time someone clicks on a link to the user's Twitter page, they're being counted. So if one of those links takes them to another website instead of directly to Twitter, they'll be counted twice. It also doesn't account for people who view part of the page but don't click through to other sites; these people won't be counted. Finally, counting unique IP addresses is not always reliable because some people use shared computers when surfing the web, so their actual view count could be higher or lower than what appears to be true from just looking at their Twitter profile.
There are other methods that can produce similar results, such as using software that tracks which accounts follow which others.
On Twitter, you cannot see who is watching your tweets. Twitter Analytics may give you information on the level of engagement with your individual tweets. However, you would not be able to see who watched your tweets, even if you used Twitter Analytics. You can only find out by checking the accounts that have been following you.
Direct engagement—a reply, a favorite, or a retweet—is the only way to know for sure if someone has seen your tweet. However, if a user wants to obtain an estimate of how many people have seen a tweet, they may go to the Twitter Analytics website. Here, they can enter a keyword or phrase and see how many people were searching on Twitter.com for topics related to their interest.
The more direct engagements you have with your audience, the more likely it is that they will see your tweet. It's also possible that someone else might see your tweet and engage with it. For example, let's say I want to tell you about a new movie that came out last week. If I post about the movie on my own Twitter page, you won't be able to read about it unless you follow me. However, if I post about the movie on another person's Twitter page, such as @MichaelCaine, then you will see my post even if you aren't following me. The more channels that we use to spread awareness about our ideas or products, the greater our reach becomes.
Finally, don't forget about Google+. Google+ posts appear in Google search results and in the "Related Searches" section of Google.com/search. Users can also find these posts if they type in Google+ directly into its search bar. Posts appear in this manner regardless of whether or not they are public or private.