If you need the current sensor to work unisolated from your measurement equipment you can very easily use a 1%, 0.5% or 0.1% shunt resistor.
The easiest circuit is measuring the return path current, from the device into the ground node. Another trick, requiring only slightly more components is high-side measurement and level conversion.
In both cases you need op-amps as well.
Be aware, that every op-amp has different characteristics and your choice of one will be influenced by many details of the current signal you are not providing.
If there is a supply voltage above the highest input and output voltages and a negative supply under the lowest input and output voltage you enter the domain of very cheap op-amps with sufficient accuracy in spades.
If you cannot provide these extended supply voltages you get stuck with rail-to-rail op-amps, these can have input and output voltages going from negative to positive supply. The basic R-R op-amp is affordable enough, but most all of them will give poor amplification performance in the extreme 500mV of the output signal, meaning fast-moving signals will be deformed and even slow-moving may have aberrations beyond your 1% specification in the last 250mV. These slow-moving aberrations are usually patterned, though, and might be solved with look-up tables in the digital domain.
Another consideration is good input performance, but I'd expect most main stream modern op-amps to be sufficient on that respect.
I attach two example schematics, one for measuring the return current, one for measuring the high-side supply current. Be aware that getting 1% accurate components will give you worse than 1% absolute accuracy, as the absolute accuracies accumulate. I'd say, use at worst 0.5% for your shunt (not cheap) and 0.1% for all the normal resistances.
Note that the example assumes you have only Vin+ and Vin- as supply voltages. Standard cheap op-amps will be better off connected to (Vin+ + 3V) and (Vin- - 3V) for the high side measurement. The return current measurement can use a cheap op-amp if the Vin+ is at least 8V and there is a supply of (Vin- - 3V) or less.
Also note, this example is "the most basic". I do not know your exact requirements so I have drawn something sturdy with no filtering, so you get the best available current representation in your output signal.
So sorry for the delay, wanted to put the picture on a new permalink private domain do "keep them in personal maintenance", but as the DNS issues take long I forgot a while. Now just inserted it directly.