Owning the Lulav Set
On the first day(s) of chag, you must "own" the lulav and etrog you wave. That means one day in Israel, and two days elsewhere. Shorthand: whichever days are celebrated as "yom tov." In theory (no longer in practice), the second day might actually be the first day because of an error in spotting the New Moon. You can borrow a set the rest of the days without issue, so long as it's kosher. Remember a) that some people have higher standards for what makes a set kosher, and b) being a plant, it can get damaged easily through use.
Where do we get this ownership rule? From Leviticus/Vayikra 23:40: "And you shall take for yourselves on the first day, the fruit of the hadar tree, date palm fronds, a branch of a braided tree, and willows of the brook, and you shall rejoice before the Lord your God for a seven day period."
But that doesn't sound right because you see that many families share one lulav set. So how does that work? Legalisms, that's how. I'm a lawyer, so you can trust me when I say that legalisms are not restricted to Judaism. That doesn't make me feel any better when I feel that a legalism is silly, so know that you're not alone.
"Borrowing" a Lulav Set
So here's what you do if you need to "borrow" a lulav. Find someone willing to let you use his set. Some people won't - I've seen it, but I can only guess at reasons. Most likely, it is a fear that you will break it and make it unusable. As we said above, they're easily damaged, and it can be hard, if not impossible, to replace them during chag.
Once you have a kind person who will give you his set, the person will "give" you the set "as a gift" (conditioned on its return). Because, you know, you might try to keep it, so we have to protect the owner against this possibility. Jewish law allows something to be a "gift" even if both parties know it should be given back. However, if you are borrowing from someone who is unfamiliar with the halacha, this doesn't work. The person must intend to "give" it to you, not just "lend" it. With just a little awkward conversation, you can make sure that you both understand what it happening. (If you are concerned that the person thinks you're "just borrowing" it, remember to be kind and tactful if you want to clarify the halacha. Don't just assume, and don't be a jerk about it. The person who doesn't know the halacha is the person you should be most kind to.)
You use "your" set, then you "gift" it back to the original owner.
Special "Borrowing" Situations
So...what if there is more than one person who needs to "borrow"? You make a chain! Easy peasy.
What if you're part of a family that has one set? Generally, because men are the ones obligated in the mitzvah of lulav, the set "belongs" to him. Women can do the mitzvah (and are encouraged to do so), but at the end of the day, he's on the hook, so it should be his. So what if the husband isn't home and someone asks the wife or an older child to borrow the set? It's valid if the "real owner" would have given it if he were there. A minor child can't give the set.
Who is the "presumed owner" in the home of a single mom? If there is a child over the age of bar mitzvah, he has a high obligation than mom. In a house of daughters and/or sons under bar mitzvah, the owner is presumably the one who purchased it.
What about roommates? I'd assume it is owned by the person who bought it, and a roommate cannot give it to another. Of course, you can always come to another arrangement.
Can you give someone else's if it's left on the seat at shul? No, don't be a jerk.
When Kids Are Involved
So...kids under bar or bat mitzvah. That's a great question too. In Israel, this isn't a problem. Since you only need to "own" the set for one day in Israel, the parents bentch lulav, then "give" the set to a kid because children should practice (this is called chinuch). The kids can share it as they wish since they are not yet obligated until bar or bat mitzvah.
Outside Israel, there's a problem because the adults need to "own" the lulav set on Day 2. The kids don't have the halachic ability to "give" the set to a parent because they can't make contracts and transfer property. That means that once a parent gives the set to a child on Day 1, the child cannot return the gift to the parent for his use on Day 2, when he must still own the set used. What does the parent do on Day 2?
A better question: why is the parent "giving" the ownership of the set to a child in the first place? Since it's chinuch and the child is not yet obligated, then the parent can lend the set. So I don't understand why this is a question in the first place, but the Gemara says it is, so it is. I have seen an argument that the child would be making a bracha in vain if the child were not "given" the set, but since the child is not obligated whether he owns it or not, I would think that the bracha is "in vain" regardless, but we say it anyway because it is chinuch. ...And this is why Rav Moshe Feinstein says the best case scenario is to buy a set for each of your children, if that is feasible. I'm not going to give you a "this is what you do" for this situation because my research shows that there is a lot going on here, and people may rule differently. So...ask your rabbi if this case applies to you. (But when in doubt, you can always get another person to "give" you his set.)
This seemed like it would be such a simple blog post, right? You know when you say a word so many times that it loses all meaning? I feel that way about "lulav" right now.