I think you should teach her how to be a friend and to get what she wants. If she wants a friend to come over, she needs to ask. Have HER call HER friend to ask to play. Not you arranging it for her. Tell the jealous daughter that she needs to speak up and if she wants a playdate, too, she needs to say so. If your daughter is in 6th grade, it may also be that her sister is simply more popular. Middle school is a tough time for girls. if her current social pool is low or snotty, then consider getting her involved in a hobby where she can meet more likeminded kids.
Even the 3rd grader can learn how to dial the phone, ask politely, etc. They should OK it with you first, of course, but they can ask friends themselves at these ages.
My SD had friends who would hang up when an adult answered the home line. They were so unsocialized they didn't know how to ask for the party they wanted and in this day of cell phones, they were just used to getting the one person they were looking for. I flat-out told SD that was dumb and she agreed.
You can also teach your kids to ask to go over there sometimes and know that (for various reasons, I was one of those kids) that not every kid can reciprocate in kind, so you may be hosting a particular child or two much more than they host your child.
When my sks were in 5th grade, they were biking to nearby friends' houses and vice versa. Yes, there is still a level of involvement with kids because they are kids, but I think that by 11, it is perfectly reasonable for the kid to say, "Hey, can Jimmy come play on Saturday?" Then you can say, "Well, we have x from 1-3 but after 3 is OK." Then your child can call Jimmy and ask if he can come over after 3 and discussions about transport can be done then. Maybe you pick up Jimmy on the way home and Jimmy's parents get him later. That's what I mean. The kids can do some of the legwork.