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.
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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.
Fighting both broadsides at once was rare, and the crew as well as the ammunition establishment assumed that in the unusual circumstance of a battle (even in wartime, a ship could go years without ever engaging), only half of the guns would be used. The reason ships had a double set of artillery was that it took too long to shift all the guns from one side to the other. This meant that it was possible to fire both broadsides once (not quite at the same time, which could put an impossible strain on the ship's structure, but ALMOST simultaneously) but, as Mr. Bird explained, the batteries on both sides could not then be served continuously. If both broadsides were being fired, reloaded, and fired again, the rate of fire would decline dramatically. This was not a specifically British problem: NO warship could carry enough men, food, and water to have a full crew for both broadsides, even if the men could have been obtained and the money could have been found to pay them. The British could get away with smaller gun crews in part because of superior metallurgy: their guns did not weigh as much as comparable pieces in most other navies, so they did not need as many men to run out.
How this worked in practice can be seen from the stations for the French 70 Neptune in the 1730s (nominally a "74" with another 4 × 8-pounders on the poop deck) [Source: Pierre Margry, "Une famille dans la marine au XVIIIe siècle, Première Partie: Les Beaussier et les dernières années de Duguay-Trouin," Revue maritime et coloniale, Vol. 62 (Oct. 1879), 231-232; and [Jean Marie] G[eorges Ferdinand] Lacour-Gayet, La marine militaire de la France sous la règne de Louis XV (Paris: Honoré Champion, 1902), 461-2, and 2d ed. (1910), 489.]:
MAIN GUN DECK (PREMIER BATTERIE), 191 men
UPPER GUN DECK (DEUXIÈME BATTERIE), 146 men
QUARTERDECK (TROISIÈME BATTERIE), 41 men
HANDLING SAILS FROM THE DECK (À LA MANŒUVRE), 79 men: Afterguard (quarterdeck), 22; waist (à la coursive), 21; forecastle, 25
MUSKETRY, 100 men: Poop (dinette), 28; quarterdeck, 30; gangways (à la coursive), 12; forecastle, 30
BELOW, 45 men
This adds up to 602 men, but does not include commissioned officers or most of the midshipmen, or specialists like the carpenter, caulker (a separate rating in the French navy), and their mates. The establishment crew for Neptune was 6 officers plus 550 men. For the same ship, 489 men were assigned for getting under way, which I will not particularize here.
A similar distribution is available for French 80 Foudroyant at the Battle of Minorca in 1756 [Source: O[nésime-Joachim] Troude, Bataille navales de la France (4 vols.; Paris: Challamel Ainé, 1867), I, 335-3]
MAIN GUN DECK (1RE BATTERIE), 243 men
UPPER GUN DECK (2E BATTERIE), 184 men
UPPER WORKS (GAILLARDS), 58 men
MUSKETRY, 135 men
WORKING THE SHIP (MANŒUVRE), 107 men
DISTRIBUTION OF POWDER, 38 men
IN THE HOLD, 22 men
BOAT (CANOT ARMÉ), 18 men
TOTAL: 805 men, again excluding commissioned officers and specialists like the maître-pilote (~ British master).