Depends what type of traiing you are talking about and what you mean by formalised.

A toolbox talk may be considered ok for some training and yet to prove competence in other reas (use of a MEWP for example) a formal course with certificate would be expected.

Well looking at the CDM acop, the Appendix on competence says something along the lines of having training arrangements to ensure your employees have the skills and understanding necessary to discharge their duties as contractors, designers or whatever.

So generally it doesn't say formal is mandatory but for me certain aspects of construction work surely must have 'formal' training, like operating a telehandler or an earthmover - this is best done by the supplier/manufacturer.

The appendix goes on to say employees are expected to have the appropriate qualifications and experience for their assigned tasks, unless they are under controlled and competent supervision.....

.... which suggests IF you are supervised you don't need formal training..... ?

It is a bit vague :o/