Charge distribution in an open circuit
Power sources don't create charge, they create voltage. Nearly equal and opposite charges already exist in every wire.

The effect of the applied voltage is to push a very small amount of net positive charge on the positive side and negative charge on the negative side. It does this by forcing a very few electrons from the positive side to the negative (these may or may not be "the same electrons" depending on  the type of source — it doesn't matter). This flow stops as soon as the force of the unbalanced charges wanting to spread out equals the force exerted by the source.

Within one side of the open circuit, the electrons will distribute themselves according to the electrostatic principles you have already learned.

If you were to disconnect the two wires from the source, then they would have a net positive and net negative charge. It's just so small (assuming the source is not supplying kilovolts) that there are typically no detectable effects.

We usually don't talk about this excess charge in wires, because it is so very small — as soon as a circuit is closed and current flows, far more electrons than that slight excess/shortage flow. But this is exactly the same thing as "parasitic" capacitance of wires, and so it can have a significant effect on the behavior of circuits — it's just not thought of in explicitly electrostatic terms.

If you make the two wires closer together and bigger — make them a better capacitor — then there will be a greater net charge on each wire/plate.