No, it is not necessary to tell your coworkers that you have given notice.
Why tell, and do I need to?
Depending on the work environment, this could create tension and negative feelings which can make for an uncomfortable last few weeks in the job.
But what if I feel that announcing is a form of courtesy?
It may be best to wait until the last 2 days before informing them of your decision; if asked directly, however, it would be wise to remain honest about giving notice without going into too much detail as to why.
Ultimately, each situation will vary depending on how close or distant one's relationship with their colleagues are - use discretion when deciding whether or not this information should be shared with others at work.