How to connect INA321 in order to measure current across a resistor?
Assuming that the 1st picture shows your test setup, I can say that input common-mode voltage is 10V which is outside the applicable range (refer to Input Common-Voltage Range under ELECTRICAL CHARACTERISTICS on page 3). Probably that's why you measure different values than expected. 

As for the circuit in the second picture, the input common-mode voltage is in the suggested range but now you should take "Input Offset Voltage" into account (refer to Input Offset Voltage under ELECTRICAL CHARACTERISTICS on page 3). Since the actual gain of the amplifier stage is 10 (defined by R1 = R2 = 100k resistors), you can measure ±2mV (±5mV max.) even if the differential input voltage is zero. Now let's look at the voltage to be measured: Since the current through shunt resistor is 250mV/1k = 250uA, the voltage across the shunt resistor -thus the differential input to the INA321- will be Vshunt = 1mR x 250uA = 250nV which is way below the typical input offset voltage. I would suggest you use higher resistance as shunt (e.g. 10R or higher) if the current to be measured is less then 10mA.

Finally, you shouldn't leave "Shutdown" pin open. Please short that pin to V+ as shown in "Fig.1 - Basic Connections" on page 10.