Is a newly engaged bride to be supposed to give an engagement gift to the groom to be?

If you are the newly engaged: Congratulations!

I say no. I, personally, have never heard of that.

In some very modern relationships, a mutual gift exchange is part of engagement along the same lines as there is no "giving away of a bride" in the ceremony and wedding costs are footed by both people, not the bride's father. For liberated types, it's a means of establishing equality in a relationship where past nuptial customs sometimes favored the man's wallet and the woman's passivity.