For the United States, units typically had their own "areas" of the battlefield and did not overlap with other units of a different corps or army unless part of a multi-division attack. Interactions could occur in the process of moving up to the line or relieving other units, however. In Normandy, members of the green 90th Infantry Division shot at 82nd Airborne paratroopers as the 90th moved up to relieve them, thinking they were Germans! If there were gaps between divisions in the line, these were often filled by members of attached units (like tank destroyer battalions or cavalry groups) or by members of one or both divisions.

In the north the gap between the 14th Cavalry Group and the 99th Infantry Division was approximately two miles across. A small party from the attached tank destroyer outfit (Company A, 820th Tank Destroyer Battalion) patrolled this opening at two-hour intervals in conjunction with an I and R Platoon from the 99th Division. There was an unoccupied strip between the right of the cavalry and the left of the 422d Infantry about one and a half miles across; the responsibility for patrolling here was given to the infantry.

Tl;dr for the United States

Infantry:

• Rifle squad: Not explicitly given in manual; essentially, "keep other squads in sight"

"Upon receipt of the platoon defense order the squad leader looks over his areas and notes the locations of adjacent squads and any supporting weapons for which he must provide close protection."

• Rifle platoon: Not to exceed 250 yards, in exceptional cases 400-500 yards

• Rifle company: Usually 500 yards, not to exceed 800-1,000 yards

• Infantry battalion: Usually 1,000-2,000 yards; as low as 1,000 to as high as 3,500 yards

• Infantry regiment: Dependent upon how many battalions are on line, essentially the sums of these battalions' frontages

Armor

• Tank battalion: Usually not over 1,800 yards in the attack

 

According to the United States Army Field Manual FM 7-10

The company commander assigns frontages in his front-line platoons in accordance with the natural defensive strength and relative importance of their defense areas. Where a platoon occupies a vital area having poor observation and poor fields of fire, such as in heavily wooded or broken terrain, the frontage of its area should not exceed 250 yards. Where the area is more open and affords longer fields of fire, a frontage of 400 to 500 yards may be assigned. Where the terrain is open and flat, or an obstacle across the front renders the area unlikely to be attacked, the frontage assigned to the platoon may exceed this figure. The area physically occupied by the platoon usually will not exceed 300 yards.

Also according to the United States Army Field Manual FM 7-10

"Where a company occupies a vital area having poor observation and poor fields of fire, such as in heavily wooded, broken terrain, its frontage should not exceed 500 yards. Where the terrain is more open and affords longer fields of fire, a frontage of 800 to 1,000 yards may be assigned. Where the terrain is open and flat, or natural obstacles across the front render the area unlikely to be attacked in strength, the frontage assigned to the company may exceed this maximum figure."

According to the United States Army Field Manual FM 7-20

"A battalion occupying a defense area on the main line of resistance will usually be assigned a frontage of 1,000 to 2,000 yards, depending on the defensive strength of the terrain. When a battalion occupies a vital area having poor observation and poor fields of fire, such as in heavily wooded, broken terrain, the frontage should not exceed 1,000 yards. When the area is more open and affords longer fields of fire, a frontage of 1,500 to 2,000 yards may be appropriate. Exceptionally, when obstacles in front of the position, such as swamps or streams, make a strong attack against an area improbable, a frontage not exceeding 3,500 yards may be assigned."

Sources:

The Ardennes: Battle of the Bulge (Chapter VII: Breakthrough at the Schnee Eifel)

http://www.ibiblio.org/hyperwar/USA/USA-E-Ardennes/USA-E-Ardennes-7.html

FM 7-10 War Department Field Manual: Infantry Company, dated 2 June 1942

http://www.ibiblio.org/hyperwar/USA/ref/FM/FM7-10/index.html

FM 7-20 War Department Field Manual: Infantry Battalion, dated 1 October 1944

http://www.ibiblio.org/hyperwar/USA/ref/FM/PDFs/FM7-20.PDF

FM 7-40 War Department Field Manual: Infantry Regiment, dated 9 February 1942

http://www.ibiblio.org/hyperwar/USA/ref/FM/PDFs/FM7-40.PDF

FM 17 War Department Field Manual: Armored Division

http://www.ibiblio.org/hyperwar/USA/ref/FM/PDFs/FM17.pdf