If they are using a employers computer to send out tweets, then yes, they should be monitored. If they are tweeting using the companies name or logo, then yes, the employee is subject to be monitored. If a employer is just monitoring their employees private life, then that is wrong. Of course, we know many companies do monitor their employees private life on Twitter and Facebook.
The only time an employer should be involved with the social media of an employee is when that employee is risking the reputation or goodwill of the company. Most private matters should be kept as such, but the honus is on the employee to be appropriately private or careful in their social media.
Employers should not monitor employees' tweets. Monitoring an employees' tweets should no considered immoral by employers. Employees' should not be judged by their social media activity any more than they would be judged for their conversations with co-workers and friends. Monitoring and employees' tweets is akin to eavesdropping or spying.
No, employers should not monitor employees' tweets, because employers should let their employees have a life. Unless the person says something about the company, they should be allowed to say what they want to say without big brother employer watching what they are doing. Most of the time, employees won't paint their employer in a bad light.
I do not think that employers should monitor employee tweets. I think that doing such a thing would be a waste of time for the employer to do. But I do agree that if an employee does tweet something that is offensive to the company he works for on a public site, the employer should have the right to fire him.