Alleged misuse of the word 'respective'

Respective designates the one-to-one relationship between the corresponding members of two different sets of things. Thus, in the examples given in the comments to your question:

• "They chatted about their respective childhoods" — A chatted about her childhood and B chatted about his childhood. One set of chatters, one set of childhoods; for each chatter there is a corresponding childhood.
• "successful in their respective fields" — A was successful in her field, and B was successful in his field. One set of successful people, one set of fields; for each successful person there is a corresponding field.

EDIT:

And, as FumbleFingers points out, if you enumerate the two sets, the members must be named in the same order.

In your example, however, there is only one set of things, languages, and there is no other set of things to which the languages stand in a one-to-one relationship.

On this question you'll find some disagreement. Despite the core meaning provided by StoneyB, sometimes 'respective' simply means 'separate, several, particular', as is attested both by the dictionary and by a corpus search.

The trouble for a lot of people is that separate, several, particular and respective are all examples of a class of 'weak' words which are often technically redundant and merely bog down sentences. (To study this argument in greater detail as applied to "respective", look here.)

Examples of 'unnecessary' respective from the corpus search link:

Their respective shares of the vote in the first round of voting were: Les verts 4.01 per cent, Génération écologie 3.62 per cent.

The trust will also recommend whether the investment costs should be passed on to consumers (a decision ultimately the responsibility of the respective regulatory bodies).

The definition in Macmillan is similar to the definition cited in the question: Macmillan

belonging separately to each of two or more people or things previously mentioned

Jane and Patrick chatted about their respective childhoods.

To re-state StoneyB, the respective items ("childhoods"; "languages", in the sentence in question) must belong separately to each individual or thing within a group, previously mentioned. In the case of “If bilingual, please list the respective languages.”, the two languages belong to the same individual. This usage is at odds with what is observed elsewhere and historically; the word is also unnecessary (the issue of unnecessary use, associated with this word, is highlighted by Merk).

To add to the answers of StoneyB and Merk:

The OED cites a number of extremely interesting obsolete senses. They provide a context on how the word usage for 'respective' has evolved, and none is similar to the usage in the sentence in the question. These senses include

1. having a reference or relation to something (1865 E. B. Pusey Eirenicon 185 But this adoration is not absolute,..but diminished, participated, and respective.)

2. appropriate (1612 T. Taylor Αρχὴν Ἁπάντων: Comm. Epist. Paul to Titus ii. 9 When occasion of speach is offred, as by questions asked, they must make respectiue answers.)

3. Of a person: attentive, heedful, careful, considerate; having regard to something. (1850 R. W. Emerson Swedenborg in Representative Men iii. 105 But Swedenborg is systematic, and respective of the world in every sentence.)

4. Of conduct, attitude, etc.: marked by care or attention, 5. Discriminating; partial (1643 Sir T. Browne Religio Medici (authorized ed.) i. §47 The day that must..reduce those seeming inequalities, and respective distributions in this world, to an equality..in the next.)

5. deserving of respect or deference; respectable (a1661 T. Fuller Worthies (1662) London 218 She was descended of a respective Family.)

6. looking or facing to a certain place, direction (1601 P. Holland tr. Pliny Hist. World I. 501 Virgill condemned altogether the planting of any trees, respective to the West [L. ad occasus].)

There is also an old usage that survived into modernity in a sense Characterized by, expressive of, or showing respect or deference; courteous, civil; respectful:

1857 M. Wheat Trav. Mexican Cordillera xxiv. 108 These dishes at table were all served in a most respective manner to the guests.

1991 Times 20 Mar. 37 A dissenting voice is every bit as legitimate if not more so than one respective of authority.

The sense that is being discussed is defined by the OED as Relating or belonging separately to each individual, group, etc., of those in question; associated with each considered in turn; corresponding; separate, several, particular, proper. Frequently with possessive adjective.

Merriam-Webster, already mentioned by Merc, gives a much shorter definition ‘particular, separate’ (MW).

However, the examples provided by MW (“their respective homes”, “it was late when the concert let out, so we all went our respective ways”, and in the section Recent Examples on the Web) suggest usage in the context outlined in the more detailed OED’s definition:

“Relating or belonging separately to each individual of those in question; associated with each considered in turn, corresponding”.

1989 M. Coren Gilbert vii. 173 A world of fighting football fans running and screaming at each other under their respective flags.

2001 Brit. Jrnl. Hist. Sci. 34 326 The respective shares that ‘Nature’ and ‘Nurture’ ordinarily contribute to the body and mind of adults.

Examples from MW’s ‘on the web’:

Darling and Rauschenberg might not agree on much, certainly not politics, and least of all their respective places in the art world. — Daniel Lee, National Review, "The Art of Parallel Purposes," 7 Sep. 2020

Each professor joining the center is teaching two classes in their respective fields and simultaneously preparing proposals for future classes in various fire specialties. — Lauren Hernández, SFChronicle.com, "New SJSU wildfire research center seeks to improve forecasting tools to help California firefighters," 7 Sep. 2020

(the section includes a number of others)