In the original X-COM trilogy, it is possible for soldiers to wield a weapon in each hand at the same time. This is called dual-wielding.
UFO & TFTD
In UFO: Enemy Unknown and X-Com: Terror from the Deep, there is no firepower benefit from dual-wielding, as firing weapons uses the same amount of TU regardless of whether a soldier is wielding two weapons or only one (i.e. you cannot fire both weapons "at the same time"). There is a direct penalty, as two-handed weapons will suffer a 20% accuracy penalty (*0.8 accuracy) if the opposing hand is not free. As such, it is generally suboptimal to dual-wield most weapons.
However, as most items can only be used when in hand, and moving objects around in the inventory costs TU, it can be useful to keep two items in hand for the sake of versatility. Usually this consists of a soldier "dual-wielding" a pistol with a flare, grenade or medikit, but wielding two weapons can occasionally be worthwhile if they are sufficiently different (e.g. Laser Pistol + Blaster Launcher in UFO, or Sonic Pistol + Vibroblade in TFTD).
In the turn-based mode of X-Com: Apocalypse, there is no TU cost for moving items around in the inventory, but there is an accuracy penalty for dual-wielding most weapons and both weapons cannot be fired simultaneously. As such, it is strictly better to put away or drop whatever is in a soldier's off-hand before firing a weapon and then pick it up afterwards. The only possible exception is if a soldier is using a Personal Cloaking Field, as holding it in hand may prevent reaction fire (even then, reaction fire is uncommon in Apocalypse).
When dual-wielding in turn-based mode, shooting a target or tile repeatedly with Shift+LMB will cause the agent to alternate between the two weapons. However the firing costs are no different than firing each weapon manually.
In stark contrast, dual-wielding is exceptionally effective in the real-time mode of Apocalypse, as both weapons can be fired simultaneously. This is obviously a straight firepower boost for pistols (as well as homing missiles, which will always hit regardless of calculated accuracy). Two-handed weapons (those with dimensions more than 2×2) suffer a 40% accuracy penalty (x1.4 inaccuracy) when dual-wielded, but this is a very good deal compared to switching fire modes (switching from Aimed to Snap, or from Snap to Auto, incurs a 100% accuracy penalty (x2 inaccuracy) for the same doubled fire rate). Of course, you do have to pay for the extra gun.
Note that dual-wielding the M4000 Machine Gun and Toxigun in Auto mode will not achieve a full doubling of firepower, as this will cap out the game's fire rate limit. It is therefore usually better to use another firing mode when dual-wielding these weapons.