NumPy random seed produces different random numbers

You're confusing RandomState with seed. Your first line constructs an object which you can then use as your random source. For example, we make

>>> rnd = np.random.RandomState(3)
>>> rnd
<mtrand.RandomState object at 0xb17e18cc>

and then

>>> rnd.choice(range(20), (5,))
array([10,  3,  8,  0, 19])
>>> rnd.choice(range(20), (5,))
array([10, 11,  9, 10,  6])
>>> rnd = np.random.RandomState(3)
>>> rnd.choice(range(20), (5,))
array([10,  3,  8,  0, 19])
>>> rnd.choice(range(20), (5,))
array([10, 11,  9, 10,  6])

[I don't understand why your idx1 and idx1S agree-- but you didn't actually post a self-contained transcript, so I suspect user error.]

If you want to affect the global state, use seed:

>>> np.random.seed(3)
>>> np.random.choice(range(20),(5,))
array([10,  3,  8,  0, 19])
>>> np.random.choice(range(20),(5,))
array([10, 11,  9, 10,  6])
>>> np.random.seed(3)
>>> np.random.choice(range(20),(5,))
array([10,  3,  8,  0, 19])
>>> np.random.choice(range(20),(5,))
array([10, 11,  9, 10,  6])

Using a specific RandomState object may seem less convenient at first, but it makes a lot of things easier when you want different entropy streams you can tune.

I think you should use RandomState class as follows:

In [21]: r=np.random.RandomState(3)

In [22]: r.choice(range(20),(5,))
Out[22]: array([10,  3,  8,  0, 19])

In [23]: r.choice(range(20),(5,))
Out[23]: array([10, 11,  9, 10,  6])

In [24]: r=np.random.RandomState(3)

In [25]: r.choice(range(20),(5,))
Out[25]: array([10,  3,  8,  0, 19])

In [26]: r.choice(range(20),(5,))
Out[26]: array([10, 11,  9, 10,  6])

Basicly, you make an instance r of the RandomState and use it further. As can be seen, re-siding produces the same results.