TL;DR There's no need for the states to coordinate directly, although they'll certainly be aware of what other states are doing. How concurrent the requests have to be is up to Congress, which has never had to make that decision.
The general way something like this would happen is that some national organization decides to push for it and starts lobbying all 50 states' legislatures. This could be the National Governors Association, in which case it would be each states' governor pushing on the legislature, but it could be any other organization. Additionally, anything like this will start being picked up by the news after a few states start voting for it, so other states' legislators will certainly know what's happening without any formal communication necessary.
There's nothing in the constitution specifying whether all the states need to call for a convention at the same time, but it certainly safe to assume that if they all did so within the same Congressional session (which is two years long), it would be considered valid. However, the 27th Amendment was ratified over two hundred years after it was originally proposed, so it's conceivable that it could be spread out over several years. If it is spread out, then it would be up to Congress as to whether they would honor the requests or not. If they don't, then it might go to the Supreme Court (which would likely find that it's a political question and send it back to Congress). Regardless of what happens, it would set the precedent for any future attempts: Either the timeframe doesn't matter or Congress will need to specify the timeframe explicitly. But there's no way to know ahead of time what that decision would be.
This link has a good summary of the issues involved in trying to organize a such a convention in the first place, as well as previous attempts (which I hadn't known about before this):
The Convention Alternative.—Because it has never successfully been invoked, the convention method of amendment is surrounded by a lengthy list of questions. When and how is a convention to be convened? Must the applications of the requisite number of States be identical or ask for substantially the same amendment or merely deal with the same subject matter? Must the requisite number of petitions be contemporaneous with each other, substantially contemporaneous, or strung out over several years? Could a convention be limited to consideration of the amendment or the subject matter which it is called to consider? These are only a few of the obvious questions and others lurk to be revealed on deeper consideration.
This method has been close to utilization several times. Only one State was lacking when the Senate finally permitted passage of an amendment providing for the direct election of Senators. Two States were lacking in a petition drive for a constitutional limitation on income tax rates. The drive for an amendment to limit the Supreme Court’s legislative apportionment decisions came within one State of the required number, and a proposal for a balanced budget amendment has been but two States short of the requisite number for some time. Arguments existed in each instance against counting all the petitions, but the political realities no doubt are that if there is an authentic national movement underlying a petitioning by two–thirds of the States there will be a response by Congress.