Oh boy, thanks for the gold!

Oh goody goody, I love this topic.

Depends on the country. Also depends on whether we're talking about "official" combat photographers or seconded civilians.

German combat photographers, for example, were members of PK (Propagandakompanie), initially a branch of the Nachrichtentruppen (communications troops), but given a separate troop branch with light grey uniforms in 1942. They furnished all official media (photo and film) for the Deutsche Wochenschau, as well as military and civilian propaganda papers. In combat, they were armed as light infantry, generally with sidearms (Walther P38 or similar). It's interesting to note that several German military branches issued their own "branded" photographic equipment (e.g. Leicas with Luftwaffe logo on them).

In the US, combat photographers could be civilians "embedded" with units, known as war correspondents (even though that term also covered reporters who were members of the armed services, such as Bill Mauldin) - Robert Capa, Margaret Bourke-White, and Joe Rosenthal are probably the best-known photographers among these, but correspondents like Ernie Pyle fell into the same category. They were officially unarmed, and counted as "civilian employees of the US War Dept.". They wore special badges with officers' uniforms (without military insignia). Robert Capa famously took along his Burberry trenchcoat to the Normandy landings, although he lost it in the water. Here is a good overview of their insignia and this is a nice collection I found online to give you an idea. They would not have been armed to avoid falling afoul of the 3rd Geneva Convention, but whether anyone got their hands on a pistol out of their own initiative is of course a matter of each individual case. Also see /u/thunderbird45's spot-on comment below.

They could also be regular enlisted or officers in signal photo companies. Combat cameramen also carried standard infantry arms when in action. The same went for the British AFPU (Army Film and Photographic Unit) which only really got rolling in 1941-1942, after only 2 (!) photographers had joined the BEF in France in 1939. Photographers were frequently unarmed, but if they did carry arms, used service revolvers (Webley, Enfield No.2 or similar). Here is a decent history of the AFPU in WWII.

Soviet photographers and filmmakers generally worked for TASS, Pravda, or one of the smaller Soviet agencies or papers, but were members of the armed forces. They were usually armed (again, light infantry weapons / sidearms). A good example is this shot of Emmanuil Evzerikhin in Stalingrad. Yevgeny Khaldei (Soviet navy lieutenant in 1945) is another well known war photographer - you probably know his (doctored) photo of soldiers raising the red banner over the Reichstag.

It's a bit pop history, but you might like Shooting War if you can ignore the idiotic added-on rat-tat-tat soundtrack.

Combat correspondents, assuming they were soldiers/sailors/airmen and not embedded civilians, were considered like any other proper member of the armed forces of that country - and thus expected to fight and be treated like any other combatant. However, their primary mission was not combat; armament was generally purely defensive. They were there to record, news (propaganda or not) was considered an important "product" in its own right. Carrying around a full size rifle, grenades, ammo, kitchen sink, you name it, would get in the way of camera handling - particularly for someone carrying 2-3 cameras, lenses, film, and maybe tripod, and expected to be ready at a moment's notice to record stuff.

Let's not forget that a very large proportion of WWII photographs were also taken by regular soldiers who'd brought their own cameras - obviously this was more prevalent in countries like the US than, say, Japan or the USSR.

Edit: If I missed anything or got anything wrong, LET ME KNOW, this is a subject that really interests me, help me learn :D

Reporters, photographers, and other civilians who followed armies during World War II were not immune to becoming prisoners of war. Article 81 of the 1929 Geneva Convention stated that, “Persons who follow the armed forces without directly belonging thereto, such as correspondents, newspaper reporters, sutlers, or contractors, who fall into the hands of the enemy, and whom the latter think fit to detain, shall be entitled to be treated as prisoners of war, provided they are in possession of an authorization from the military authorities of the armed forces which they were following.” Hope that sheds light on the treatment of correspondents during the war.

Article 81, Convention Relative to the Treatment of Prisoners of War. Geneva, 27 July 1929.

While answering a question about iconic war photographs in /r/history, I came across Ernest Brooks, who took some of the more famous shots. The National Library of Scotland website notes:

Ernest Brooks was the first British official war photographer to be assigned to the Western Front in 1916. Previously a 'Daily Mirror' photographer, he was given the honorary rank of Second Lieutenant. His remit was to take as many photographs as possible, with as much variety as possible. Using his inconspicuous hand-held camera Brooks was free to wander, sometimes capturing his subjects unawares. Many of the images taken by Brooks were used to fuel the propaganda machine at home and abroad. Despite this Brooks, who was very aware of composition and light, produced some very artistic and thought-provoking images.

Other photographers listed on the same site were in similar situations: civilian press photographers were given army commissions to give them access, in exchange for which they produced images of propaganda value. You might think photos of the dreadful conditions at the front would be forbidden, but they seem to have been used to illustrate the Good Old British Grit Of Our Brave Tommies.

Individual soldiers were in theory prohibited from having cameras and taking photos for obvious security reasons, but many took compact cameras with them and recorded everyday life.

My grandfather was part of the Canadian Army Film and Photo Unit, so this topic is very close to my heart! You might be interested in a documentary about this unit called "Shooters" (http://www.jamesoregan.com/jamesoregan/Shooters.html), which includes interviews with many of the surviving photographers.