Does 'each other' function as singular or plural?

If you notice that each other, while idiomatic, is not unbreakable, it begins to come clear.

The reciprocal phrase each other can be separated into one determiner binding something in the subject (each, each one), and one determiner binding something in the object (other, the other), viz:

• Each (one) of us knows what the other (one (of us)) is doing. (dual)
• Each (one) of us knows what the others/other ones are doing. (plural)

And some of these quantifiers may be floated to pre-verbal position:

• We each know what the other (one) is doing. (dual)
• We each know what the others/other ones are doing. (plural)

From this it's easy to see how each and other hook up:

• Each of us knows the other.
• We each know the other.
• We know each other.

But that doesn't resolve the number problem that comes from having two determiners potentially conflicting. So both

• *We know what each other is doing. and
• *We know what each other are doing.

feel wrong.

Of course, there are other quantifiers that specify more precisely,
especially when they're separated, viz:

• We both know what the other is doing. (dual)
• We all know what the others are doing. (plural)

"Each other" refers to a singular subject, just like "one another" does. Notice "each" and "one" are singular; so are "other" (an other v. others) and "another" (which is like a contraction or compression of "an other").

Another way to think of it is: "we know what each other is doing" means "I know what he is doing and he knows what I am doing"...all singular.

No, neither needs to be correct because there is a problem of verb agreement. "We" means you and I (inclusively, anyway), so instead of "is doing", you'd need "are doing" for "you" and "am doing" for "I". There is no such problem with "They know what each other is doing" (which, however, still seems a bit odd).