Is there any data to support or refute the hypothesis that sailing ships of the line were only complemented with enough gunnery crews to simultaneously fire 1 broadside but not 2?
If it matters for precision, let's assume British ships of 1700s.
For British ships of war, the usual practise was to have a single gun crew allocated to the guns on both sides of the ship. If the ship was fighting on a single side then the whole gun crew worked the gun on that side. For the great guns this was between 10-14 men depending on the size of the cannon.
If fighting both sides, the gun crew split so that a smaller number of men worked both guns. The ideal would be to have the guns working out of sequence so that one was being run out while the other was being sponged, wormed and loaded. This allowed men to swap between guns as they were needed (in effect having a full team for each gun), rather than having half the team working one gun and half the other.
As casualties were expected in battle, the best trained gun crews rotated their positions so that each man, if not expert, was capable of performing in any position in the crew. That way the loss of any single man would not cripple the gun.
ref: Shipboard Life and Organisation 1731-1815, NRS vol 178 (1998), ed. B. Lavery. pg 274
With regard to the idea that ships of the line were "understaffed", it's worth bearing in mind that, in most cases, the gun crews made up 80-90% of the ship's crew. Also these were ships that, by modern standards, would be considered uncomfortably overcrowded as they were.
Therefore, if you wanted to have a full crew per gun, you'd almost double the crew. Not only would you have to find accommodation for those extra men, and carry extra supplies of food and water, you'd also have to find jobs for them to do for the long periods when they weren't in battle or drilling on the guns. The benefit of the extra men would only be felt in a battle when both sides of the ship were engaged.
Finally, to address a comment on another answer, the ship's marines very often provided men for the guns. I believe that, traditionally, they manned the guns and carronades on the quarterdeck. However in some cases, such as on board the 74-gun HMS Goliath, they were used throughout the ship. The quarter bill (which shows the stations of every man in battle) for the Goliath, shows that every gun crew on the main and lower decks had 2 or 3 marines assigned to them.
ref: Shipboard Life and Organisation 1731-1815, NRS vol 178 (1998), ed. B. Lavery. pg 276-287
The crew losses during broadside actions were so high that even if the ship of the line was able to man both sides at the beginning of the battle, its crew would probably be depleted an hour or two later.
Also, even if there were not battles, ships had crew attrition from just being at sea (scurvy, accidents). A ship of the line could hold max. one thousand man crew (often this number was closer to 500), with approx. 10 men / big gun you'd need 720-1000 men to man all guns (three deckers had at least 72 guns) -- who'd be left to handle the sailing duties? Also this number included the marine component, which was 20% of the crew for British and up to 30% for Spanish warships.