Privacy has emerged as an interesting concept in today's digital age. The Internet, born masked underneath the cloak of anonymity, has evolved to become an open area of sharing, which the society at large comes to congregate, communicate and distribute information.
Today's socially centric web has changed the dynamics in how people interact and share, and the whole foundation of social media is rooted in sharing. And a lot of it is done using real names and identities.
Anything that is displayed on the web opens up some risk of privacy, and Twitter is not excluded. People often create user names, but these are also often connected to real names these days. Credibility is necessary, especially for those who are on Twitter for marketing, a social cause, or other business purposes, and this includes using a real name, and often, a photo.
Personal information that is placed in profiles begins the unraveling of privacy, however, other factors contribute too. There are many ways privacy can be exposed on Twitter:
Probably the largest form of privacy risk comes from the users themselves. Members often tweet about where they are, what they are doing, where they work, and more. Through these tweets they expose an awful lot of information about themselves either immediately or over time. Whether or not privacy is desired by those users is another issue. Most people nowadays understand that by engaging in social media websites, such as Twitter, it means a degree of privacy is given up. There are many people who don't mind sharing. However, some people share too much and put their privacy, and security, at risk.
Spilling too much information on Twitter (or any social network really) can lead to privacy issues even if a user thinks he or she is being careful.
Aggregation of Tweets
Unless privacy restrictions are placed on accounts, and many members do not use these, all tweets are out for public display, and can show up in search engines or other websites that include Twitter streams on their pages. Put together it can create a fully intact puzzle about someone's lifestyle, professional life and other habits, on or offline. These profiles extend beyond the Twitter network with widespread sharing. Privacy settings can be used, but when looking to gain audiences, this isn’t exactly prudent. It might be better to set up separate accounts, one private for personal and one public for business.
Websites can be hacked, users can compromise their own privacy or other mishaps can happen that result in breaches of data. This is true of any company and Twitter is no exception. Any website is at risk for a breach and, if it occurs, whatever information you've shared internally on the network can be exposed. This, however, is no more of a risk than sharing with any other business, however. Although, the larger the membership grows, the more likely Twitter may be a target to those with sinister intentions. In fact, the social network fell victim to a massive breach in June 2016. 
Twitter said its own servers were not breached, but that malware and breaches of other websites led to 33 million of its own members getting breached. Many of those breached were found to have reused passwords on several websites. (Another good reason to always use unique passwords!)
Third Party Apps
Many Twitter members routinely use third party applications to help customize and/or enhance the Twitter experience. Unfortunately, the more hands in the proverbial pot, the higher potential privacy can be compromised. Any site can be compromised which could have a ripple effect. Additionally, some apps may not be legitimate ones, further putting both privacy and security at risk. 
It is not uncommon to hear nowadays that people get reprimanded, or even fired, for comments made on social networks. This is especially true if Twitter connections are with bosses and colleagues, however, tweets can be seen by friends of friends, so this can put you at risk. Oftentimes other websites may pick up on tweets and retweet them, and then your name be shown in search engine results as well.
PC World reported:
"Saying something as obvious and seemingly innocent as "I'm bored" in a status update during work hours can have dire consequences if the wrong people see it. With services like Twitter, or the recent changes to Facebook that allow anyone to view and search updates, you really have no way to hide." 
Higher Insurance Rates?
Some outlets are speculating that Twitter [and Facebook] users might put themselves in a higher insurance risk category, increasing their rates. Yale Law Tech reports:
"Why would social media users see home insurance hikes? Because the status-updates and other information they furnish on the social media services they use, such as Twitter or Foursquare, alert burglars as to when they’re home, and when they’re not." 
While this is not a firm policy, past trends have shown that commercial business has taken a huge interest in what's happening on social media sites, whether it be employers or other companies. Twitter is definitely included in that category. How tweets impact tomorrow's cultural norms is a distinct possibility and a trend many are watching.
Any level of socialization reduces privacy, however, how much privacy is lost may or may not be controlled by the user. Most importantly, knowing the risks and subsequently learning how to mitigate them is of value for any Twitter user. If privacy is of importance, users should always be careful what information they share from their accounts, be vigilant about what is posted and always carefully select what third party applications they allow access to their accounts.
Related Reading: How Sharing Information on Facebook Can Lead to Identity Theft