@Eka LM358 and LM324 are single supply opamps, they don't need + and - supply. It's a bit tricky when the output gets near the negative rail - the closer you get to it the lower the current sink capability (I didn't know that). That's what I didn't take into account. If I had -1V for the negative supply it would be fine. But I don't have. Even though I'm aiming for low price (LM324 costs 1/5 of the price of a current monitor) I would use current monitor if it was available locally. I've ordered few and while I'm waiting for them I'm trying to make it work with the opamp.
Negative supply can be done with charge pump, but they are also not that cheap. Of course for the two current senses I need one charge pump, so it's better than two current monitors.
I know why it works at the low side. When there is near 0 volts on the Vin- the opamp does not have to sink few mA and it can get closer to the negative supply.
I'm thinking something like (the output goes to ADC, max output resistance should be 10k, so this should be ok, adc input capacitance is 50 pF):
... I don't have much experience with source follower. I read in wiki that output impedance is ~ 1/g
m (transconductance) which is min 80, typ 320 mS for 2N7002. So 1/80mS = 12.5 Ohms. Should be a problem driving 50pF. Even though capacitive loads of the transistor and the ADC are not big I've added series resistance and integrating capacitor. Maybe the capacitor can be smaller, but I don't need that much speed and I don' know how to check for stability.