Will the cations flow to the right side of the beaker, while the anions flow to the left side?
To some minimal extent yes. However, try to calculate charge separation necessary to neutralize electric field in a beaker - assume it is enough that all 'separated' ions are on the beaker surface. Even in relatively high voltages these charges are pretty small.
You may also read about charged double layer and its capacitance - this is another example of charge separation in the solution (although this is a specific case, as we are talking about ordered part of the solution).
While in general solutions are electrically neutral, they are neutral only statistically - if you take solution of NaCl and split it into parts (just pouring it down, so that it separates into droplets on the fly) there is no process that guarantees every drop is perfectly neutral. Rather, some will have small positive charge, some will have small negative charge. Think about it this way: imagine water sample containing exactly one Na
+ and exactly one Cl
- - it is electrically neutral. Ignoring for a moment water autodissociation, if you split this sample into half, it may happen that both ions land in the same half, and both half are still neutral, but it can also happen that each ion lands in different half, and neither is neutral. If you have more NaCl dissolved situation is identical - after splitting solution there is no guarantee all ions will be distributed evenly. However, they will be distributed pretty close to 50/50. Finding an average charge makes an interesting probability question.