First you cannot tell if the entangled partner has been measured, this would lead to faster than light signaling which should not be allowed by relativity.
Secondly, if you only measure both qubits in the $|0\rangle,|1\rangle$ basis you cannot say anything about entanglement. The particles could be also in a mixed state.
In order to truly test that there is some weird quantum phenomena going on you need many ensembles of the pair of entangled particles and measure in a least three different bases. For flying photons or electrons this can be achieved by just rotating the angle of one of the polarization measuring devices. See Bell test for more information. If the measured correlations between the different bases violate Bell's inequality (as entangled particles do), it leads to the conclusion that some of the following are true: devices are able to modify hidden parameters that are not localized with the particles (nonlocality), particles do not have defined quantities until one of them is measured (nonrealism), the devices are always highly correlated with the particles in order to reproduce the results of quantum mechanics (superdeterminism).
In Copenhagen interpretation, we just say the particles follow Schrödinger equation until you measure one of the entangled particles, in that moment you collapse the state of both. This reproduces very well the experimental results.