It will not trigger anything in itself unless there is some issue over the tax which may be brought up by HMRC in dealings with the employer.

The most likely sign to an employer that one has other employment are changes to the tax code such as where bennefits in kind are received through the self employment.

Tax matters where one is self employed or a director of a limited company are dealt with through self assessment and are not available to one's employer. However, you mention dividends and if one were a director then your status would be available to view on the companies house website if an employer instigated a search.

The question is not whether the employer finds out but rather whether there is any reason why the employee should not have another job.

Certain contracts of employment restrict working in a similar trade or working within a proximity of the first business.

One can argue about restriction of trade until the cows come home but the reality is that the employer has employed someone on the basis of their availability and commitment and if that is not the case then there may be grounds for dismissal.... As well as grounds for unfair dismissal. All very messy.

All in all, if there is no reason why the person in question should not have another business and they go nowhere near the clients of the employer or seek clients that might have gone to the employer then what is the problem with the employer finding out. And if there is a reason such as going against the contract of employment or stealing clients then all that they are doing is awaiting the innevitable (eventual) long walk to the managers office.