(Standard disclaimer: I am not your lawyer. I am not here to help you.)
You've tagged this "labor law," but as I understand it a big part of this question is about international contract disputes rather than employment law. I'm not a labor law expert by any means; if there's some special laws governing international employment agreements, I don't know about them.
Let's hypothesize that A, a United States citizen, seeks out, makes an agreement with, and pays B, a national of some foreign country, to do some task. B, sitting at home, tries to do it, but A is dissatisfied and wants her money back. B refuses and issues that famous ultimatum: "So sue me." What can A do?
(It doesn't necessarily matter who's the "employee" and who's the "employer" for the jurisdictional issue I'm talking about; it would work the same if B, a foreign resident, hired A, a U.S. resident, to do something but didn't pay. The important part is that A is the injured party.)
There are at least two fundamental issues here: (1) what court(s) will have authority to impose a judgment binding a breaching party; and (2) what law will apply to the relationship. The first issue is, in Anglo-American jurisprudence, generally called "personal jurisdiction"; the second is called "choice of law." (There's a separate issue, called "subject matter jurisdiction," that would probably arise in real life but that's beyond the scope of this question.)
A would prefer to sue B in her local United States court; let's say in California. Under U.S. law, a court must have personal jurisdiction over the defendant--B--in order to hear the suit. The outer bounds of that jurisdiction are spelled out in a series of famous Supreme Court cases, which held that a defendant must have done something in, or done something aimed at, the forum state. (Maybe in the future we'll have a standard introduction to minimum contacts jurisdiction on Law.SE; for now, Wikipedia's article isn't terrible.)
In my hypo, B didn't do anything at all regarding California: he was sitting in his home country until A sought him out. In that case, a California court could not exercise jurisdiction, and A would be out of luck.
In real life, the services you're talking about probably do seek out California and other U.S. clients. They run advertisements aimed at California residents, they build relationships with California residents. That may or may not be enough to open them up to suit in the U.S. (With a "marketplace" site like Elance, this is a bit more complicated because there are three people involved: A, B, and Elance.) This isn't a very clear area of U.S. law; different courts have approached online transactions in different ways. I'd have to do some research to see what the relevant decisions are.
Even if A got a judgment in a U.S. court, she may not be able to enforce it in B's home country. This would depend entirely on the laws of that country, and on treaty relationships between the countries.
A's other option, of course, is to sue B in B's home country. The procedural laws of that country would apply. If things work there like they do in the United States, you can basically always sue somebody in the court where they live, are headquartered, or are incorporated. That court would apply its own choice-of-laws rules in determining whether to apply U.S. or local law to the contract.
Now in reality, there's a good chance much of this problem, as well as the choice-of-laws question I haven't talked about, will be resolved by agreement between the parties. If I were A's lawyer, I'd make sure that the contract included a "forum selection clause" specifying that any lawsuit would be tried in a California court, and that B agreed to the jurisdiction of that court. While that may not totally resolve the problem, it'd go a long way. This isn't exactly the answer to your second question. I'm talking about B's agreeing to be sued in A's jurisdiction. B may or may not still be able to sue A in B's home jurisdiction; that's a different question.
You mentioned elance.com; their Terms of Service page includes the "Independent Contractor Services Agreement" that they provide as a default agreement between their clients and freelancers, as well as the "User Agreement" that applies to everyone using their site. The user agreement has a section entitled "Law and Forum for Disputes," which sounds like a forum selection clause, but it's not; it just says that the agreement will be governed by Delaware law, but doesn't require consent to be sued there. The mandatory arbitration clause explicitly doesn't apply to foreign freelancers. I don't know why they chose not to require consent to be sued in Delaware; there may be a good reason. But on brief review, this particular contract doesn't necessarily help a United States plaintiff as much as it could.