Your coin example is analogous to only one specific state of entanglement. That is referred to as Bell's state. This is a state of perfect anti-correlation. I will only talk about this state. Taking an example of spin - if the two particles of entangled pair are measured along same axis, then they will be always found to be opposite spin. Another way to say "always opposite spin" is perfectly anti correlated.
Perfect anti correlation can be easily explained in terms of your example. You do not even need to shoot a coin, you may just separate a pair of shoes, or socks, or gloves! Perfect anti correlation is a consequence of conservation laws. In case of spin, if one particle gets clockwise spin along X axis, then the other has to get anti-clock wise spin along same axis, in order to conserve angular momentum.
Next level of complexity is that you measure them along any axis, and you will see they have opposite spin. This is complex, but still explainable in terms of conservation laws. Specifically because, there is no measurement at quantum level, it is an alignment which is called "measurement".
Above two, in isolation, can be explained in terms of local hidden variables. Because, the above two involve only one pair at a time for checking the anti correlation.
But there is another level of complexity - Video suggested by Aaron Stevens is one of the best.
Third level of complexity is the one that baffles most people. It is statistical correlation. By definition, statistical correlation, has to involve many pairs.
Suppose you measure one particle of the pair along X axis and the other particle along an axis that is at theta degrees to X axis. There are only two possibilities - 1. they show same spin, 2. they show opposite spins. If they show same spin, let us say they are correlated.
Now you repeat same as above on 2nd, 3rd, .... n pairs. For a very large n, the number of correlated pairs is predicted by quantum mechanics as a formula which I will not write here.
All experiments show that the QM formula is correct.
Then there is a Bell song that you will hear a lot, almost from everyone. If you pay attention to the video, Bell considered all possible outcomes equally likely and came up with an inequality, which obviously, is violated in all experiments.
Bell's theorem as mathematics is just fine. But the issue with applying Bell's inequality to statistical correlations of entangled particles is - All the possible outcomes are not necessarily equally likely.
Even in a process that is as simple as hitting a mountain continuously with a hammer, your past hits have an impact on outcome of subsequent hits.
So having generated and aligned numerous pairs, we can not say the possible outcomes for the next pair are equally likely. And this fact renders application of Bell's inequality to entanglement - void.
So finally, entanglement is baffling, but has been mathematically formulated pretty well. Only thing missing is some realistic explanation which I think will turn out to be much simpler than it seems. Nature is simple at the core.