Normally, reciprocals take a plural noun (or several coordinated noun phrases) and make the entities that make up an agent (or comparable role) act upon each other.
A rather interesting situation is the use of collective nouns and reciprocals:
?the team watched each other in astonishment
*laget såg förvånat på var-andra (literally: team-the saw surprisedly on each-other)
Yeah, no, the ? mark there is probably wrong, for a huge majority it probably genuinely is *. I am convinced that some languages permit, without hesitation, constructions analogous to the one above.
Historically, "one another" and the Russian construction "drug druga" (drug = nom, druga = acc) probably consist of two bits - one ("one", "drug") encoding the subject, and the other ("another", "druga") the object. From a purely abstract look at it, it feels like this construction might be marginally more tolerable in English? In Russian, I have it on a fairly reliable source that this is entirely acceptable.
*the team watched one another in astonishment
para obnimala drug druga
In Swedish, some passive verbs primarily have a reciprocal meaning:
vi kysstes ('we were kissing')
vi slogs ('we were fighting (each other, but can also mean 'we fought random people')
Now, most Swedes accept
paret kysstes ('the couple were kissing')
despite this being a reciprocal with singular subject.
However, I can't just let it stand at "this is okay in a language so there you go", can I?
We could imagine, for instance, that a language does get any number of congruence or discongruence-phenomena with this.
1) Singular forms of 'each other'.
In Swedish at least, 'varandra' is morphologically plural. One could imagine a system where a singular form appears with a singular subject. In Russian, 'drug druga' is morphologically singular, but one could imagine a partitive genitive instead on the subject noun - getting us something like 'druga druga'.
Here, an unrelated idea appears: in the dual, a language could very well have a reciprocal singular pronoun: 'both saw the other', but 'the thief and the policemen saw each other' .
2) Plural verb forms and adjectives with singular subjects.
Some varieties of English permit plural verb forms with collective singulars, but one could imagine a situation where plural verb forms only are used with singulars in situations where the singular is blocked by the presence of a reciprocal pronoun.
3) Congruentially Forced Plurals
The subject could exhibit other behaviors that are typical of plurals, including morphological or congruence-related behaviors. Maybe the reciprocality forces an explicit plural marking, so that "the groups saw each other" can signify both 'the members of the group saw each other' or 'the different groups' members saw the other groups'. Maybe adjectives need plural congruence. Maybe they are forced to take a plural article.