Should companies let their employees choose the training they'd like to undertake or should the company assign it for them?

In most cases, an organization is better suited to understand and recommend training needs for employees rather than the employees themselves. In my decade long experience, I have seen that the moment we ask employees to suggest trainings they "need", all we get is trainings they "want" and there is a huge difference between both!

If we talk about employees, training needed for business need, project need or for any vertical, company must assign it for them accordingaly. 

Whatever training or education needed to get a job, employee had it that's why she/he became employee.