No that is not how quantum entanglement works. Here's the standard example of how it works:

Electrons have a property called "spin." You can measure the spin and it will have a value of either "up" or "down."

The spin is in an indeterminate state until you measure it. Then when you measure, it will be either up or down, randomly. If two particles are entangled, then if you measure one and find that it's up, you already know that the other one is down.

But if you're that second person and haven't been tipped off by the first person, then you still just have an electron with a spin you don't know. So you'll measure it and find out that it happens to be down. Now you know the other one is up. But it's still just a random value.

Quantum entanglement does not cause particles to move around like you described.