I have the following code:

import random

rand1 = random.Random()
rand2 = random.Random()

rand1.seed(0)
rand2.seed(0)

rand1.jumpahead(1)
rand2.jumpahead(2)

x = [rand1.random() for _ in range(0,5)]
y = [rand2.random() for _ in range(0,5)]

According to the documentation of jumpahead() function I expected x and y to be (pseudo)independent sequences. But the output that I get is:

x: [0.038378463064751012, 0.79353887395667977, 0.13619161852307016, 0.82978789012683285, 0.44296031215986331]

y: [0.98374801970498793, 0.79353887395667977, 0.13619161852307016, 0.82978789012683285, 0.44296031215986331]

If you notice, the 2nd-5th numbers are same. This happens each time I run the code.

Am I missing something here?

rand1.seed(0)
rand2.seed(0)

You initialize them with the same values so you get the same (non-)randomness. Use some value like the current unix timestamp to seed it and you will get better values. But note that if you initialize two RNGs at the same time with the current time though, you will get the same "random" values from them of course.

Update: Just noticed the jumpahead() stuff: Have a look at How should I use random.jumpahead in Python - it seems to answer your question.