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Take the example of a copper and zinc galvanic cell, connected by a salt bridge of $\ce{KNO3}$. I understand how the reactions will result in positive and negative charges, and that the ions of the salt will migrate to maintain neutrality. But why is the salt needed in the first place?

In one beaker we will have an excess of zinc ions (resulting in positive charge) and the other with an excess of copper ions (resulting in negative charge). If we just have a bridge, perhaps just soaked in water, wouldn't the ions, if not purposefully then randomly, travel across the bridge and maintain neutrality? What is the purpose of the salt then, if it works without a salt?

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The purpose of the salt bridge is to prevent the two half-cell solutions from mixing. It is possible to make a really bad galvanic cell by putting both half-cells in a single solution, but they rapidly self discharge as the oxidizing agent ($\ce{Cu^+}$ in your example) can diffuse through the solution and react directly with the other electrode (Zn in the example), with the electrons being transferred directly instead of going through the external circuit.

Having a "bridge" of just water soaked paper is similar to having both half-cells in the same container but poses some issues. First, the diffusion of ions through the bridge is going to be very slow, increasing the cell resistance, second, the self-discharge problem above will occur, and third, because different ions diffuse at different rates, a junction potential develops as the two containers diffuse into one another.

So a salt bridge's job is to not only maintain electroneutrality between the two containers, but to do so without allowing the solutions to mix. An inert salt is used to prevent any reactions from occurring between the salt and the solutions, and things like $\ce{KCl}$ are commonly used for analytical cells as potassium and chloride ions have very similar diffusion coefficients, minimizing junction potential.

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    $\begingroup$ But even with a salt bridge don't the two solutions mix anyway?(at a small rate) does the salt impede this mixing? $\endgroup$
    – QCD_IS_GOOD
    Commented Nov 16, 2014 at 20:28
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    $\begingroup$ Sure, if you leave it long enough. Often a salt solution is turned into an agarose gel to fill the bridge or glass frits placed at either end of the tube to limit the rate of diffusion. $\endgroup$
    – Michael DM Dryden
    Commented Nov 16, 2014 at 20:31
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    $\begingroup$ Why do the ions in the salt bridge have to diffuse into the half cell? Could we plug up the bridge completely and just let the ions electrically neutralize the cell by moving into the cell volume while staying inside the salt bridge tube without diffusing into the cell solution? $\endgroup$
    – yippy_yay
    Commented Mar 31, 2019 at 15:19
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This is an interesting question with two possible answers that I can see. If you are talking in terms of a galvanic cell used as a source of electrical energy, the current supplied by the cell has to be matched by the current flowing between the two half cells. As pure water is very nearly a perfect insulator due to its extremely low degree of ionisation, the internal resistance of the cell would make it impossible to supply any useful quantity of electical power. The use of an ionic solution overcomes this problem and the higher the concentration of ions the better, hence saturated solutions. However, if your question is referring to cells for measuring electrode potentials, I am not so sure of the answer. In principle pure water should do, as you suggested because its slight degree of ionisation means that it does conduct electricity very slightly and this should be all that is needed since electrod potentials should be measured when no current is flowing. I am only sorry that I did not think of this question when I had access to the apparatus needed to try it out.

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In fact, it is possible if only water and some gelatin or agar-agar is present in the salt bridge. But for $\ce{SO4^{2-}}$ Or $\ce{Zn^{2+}}$ to travel through salt bridge it will have to face a lot of resistance resulting in low voltage. Instead, it is better to have strong electrolytic salt which can maintain neutrality of solution as soon as zinc atom is oxidized or $\ce{Cu}$ atom is reduced.

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