If you haven't yet met the parents of your daughter or son's fiancé(e), that meeting should be arranged as soon as possible after the engagement takes place. It doesn't matter who takes the first step. When it comes to meeting your child's future in-laws, don't get hung up on "who should call whom first."
While the tradition holds that the groom's parents should make the first contact, these days it doesn't particularly matter who makes the first move (although the bride's parents may want to wait a few days, to give the parents of the groom a chance to honor custom). The important thing is for the parents to meet, even if over the phone first, and share in the spirit of excitement over the future union.
The actual invitation can be extended in whatever manner is convenient. While a written note is always nice, a phone call or e-mail is also fine. The point is to introduce yourself to your son or daughter's future in-laws and share your pleasure over the engagement, then go on to explore the possibility of when you can sit down together. Let the spirit of friendship be your guide: If the distance prevents you from getting together before the wedding itself, then both sets of parents should make an effort to stay in touch by calling or writing in the months prior to the big event.
Who actually hosts the get-together and the nature of the occasion are matters of preference rather than tradition. This may be a good time to consult the bride- and groom-to-be, since they're likely to know what kind of a gathering will put everyone at ease: a casual barbecue or a weekend supper, a weekend dinner or a brunch at a restaurant.