What percentage an employee engagement budget should be in relation to compensation budget or a training budget?

In my opinion, there is no actual connection between employee engagement and the compensation/ training budget.
Perhaps you are referring to something like an entertainment budget /wellness budget that should be budgeted for at the beginning of the year and the percentage for this depends on the company.
This could be allocated for social gatherings, team activities, etc.
Employee engagement is at its highest when employees understand the big picture; they know they are compensated fairly, recognized for top performance, have ample opportunities for training & development and last but definitely not the least, a great environment to work in.

I agree with Shobha, Employee engagement and training are not related; yet it should be budgeted under the social affairs. Its value will depend from organisation to another and even as Anita said; it can be achieved more economically with very marginal cost in some cases although still it depend on the context.

I believe that the Employee Engagement and Training budgets must both be equal but these budgets must be lesser than the Compensation budget.
It would be advisable to have monthly employee engagement activities - many companies divide these activities based on the number of employees E.g - a section of a manufacturing company has500 people - either an activity can be arranged for these500 separately or it can be arranged across a period of time for smaller groups - keeping in mind business continuity.
E.g Family days when the Families of the employees can visit the office and see where the employees work and what they do - lunch to be arranged for the family and employees.
Picnics and excursions to help the employees have a bonding experience out of the office premises.

Employee engagement is not requiring any budget.
You don't get engaged staff by organizing events or dinners.
It's the things that doesn't cost a thing.

There are a few points:

Training budget for an Organizations is higher

Compensation budget is allocated based on the employees Rank & performance

Employee engagement budeget is a variable allocation budget for any Organization, because employee engagement activity is an strategic proportion of different seen & unseen factors within an Organization with employees.  It is decided by the management to fill the GAP in-between employees satisfaction level and also this acts like a bridge in-between an Organization & Management.

So in my Openion everything has its own budget allocation and requirement.