For the vast majority of people there is no solid reason for not telling your current employer or coworkers.
If there were some sort of agreement between the new and the old organization to not poach employees, the new employer would have told you about it. If they didn't they would soon realize that they were wasting their time and money interviewing employees from your company.
If the current boss calls the new company and tries to sabotage your new position, that tells you that your current employer is not somebody you want to keep working for. Plus it opens the current company to a lawsuit. It also a bad sign if the new employer drops you based on a phone call from the one person has a conflict of interest regarding judging your ability to perform for the new company.
Now you have to evaluate your individual situation, you have to decide how to inform people and not sound like bragging. You have to be sensitive to your current company and their workers when telling them you are leaving, and where you are going. Also don't resign until the paperwork and background investigations have been completed.
Not telling your coworkers doesn't shield you from them trying to join you at the new company, unless you plan on dropping out of social media, and LinkedIn.