People are fleeing in both directions. One could expect that those with more pro-Russian attitudes are more likely to head to Russia while those with more pro-Ukranian attitudes are more likely to head to government-held territory, but there’s other factors at hand such as resources individual families have in each possible area to flee to. Also it should be noted that a lot of the pro-Ukranian side has long ago fled rebel-held areas out of fear of retribution.
Where the civilians go is mostly a practical decision. Those with family or a place to stay in Russia will go there. The “refugee camps” built in Russia are empty or being used by Russian combatants.