Disposal at sea and open pit burning (both of which were practiced by the British and the Soviets) are now prohibited by the Chemical Weapons Convention. So they won't be dropping barrels of anything over the sides of the ship, or simply pouring the weapons into a fire. They have to be broken down carefully so that what remains is much less dangerous than what they started with.
The weapons are mostly organic compounds, so the bulk of the waste is hydrogen, oxygen, nitrogen, and carbon. Some weapons have used arsenic, chlorine, fluorine, phosphorous, and other elements. Initially they were disposed of by single stage incineration, but that produced smoke that was toxic in its own right. Modern disposal techniques use multiple stages of heat, filtration, oxidation, electrolysis, or even detonation of the weapons. Bleach is an effective oxidizer, but gives off chlorine. Hydrogen peroxide and high temperature steam will also break down many of the compounds. Sarin [2-(fluoromethyl phosphoryl)oxypropane], which the Syrians are accused of using in this war, can be broken down by a low temperature burn, followed by a scrubbing process, followed by a second high temperature burn. Any solid materials remaining after the scrubbing and filtration processes are then buried.
A UK firm was having great success using electrolysis of silver nitrate in a nitric acid solution, which was very effective at breaking up and oxidizing the compounds, but they have ceased that research. The French are building a detonation chamber, where the entire warhead is placed in a blast chamber and detonated. The high pressures and temperatures perform an initial breakdown of the agents that destroys most of the chemicals, and the remaining chemicals are treated through an incineration and filtration process. One advantage is that it destroys both the chemical weapon and the explosive, meaning they don't have to have a separate hazardous process to handle the high explosives (which might be in a less-than-safe state of stability.)
Destroying the Poisons of War, by the Royal Society of Chemistry, is an interesting read on the history of the topic.
And the U.S. Army, who is in the process of decommissioning their Utah disposal facility at Tooele, has been drilling test holes in their disposal plant and sampling the soil beneath. So far, they have found no traces of any of the weapons they had been destroying. This plant was originally commissioned in the 1970s, so the engineering, the chemistry, and the processes have proven highly successful at safely destroying the US stockpiles over the long term. They know how to do it.