Working with 1 melon at a time, cut in 1/2. Gently scrape out seeds and membrane and discard. Using a melon baller, scoop out rounds from flesh and place in large bowl. Discard skin.
Keep the prosciutto loosely covered with plastic wrap to prevent them from drying out as you start to assemble 1 purse at a time. Place 1 melon ball in the center of a prosciutto slice, gather all sides to the top and secure with a blanched chive. It may be necessary to overlap a few pieces of prosciutto in order to completely wrap the melon ball. Place finished purse, seam side down onto a baking sheet and keep covered. Repeat with remaining prosciutto and melons. Can make it only up to 2 hours ahead of time because the salt in the prosciutto will break down the melon.
Scatter the chives onto a decorative platter. Place prosciutto purses on top and serve.