As someone who published a history book in the USA recently that is being translated (into Chinese!) — my contract with the publisher basically says that they are in charge of negotiating translation rights. So, in my case, they went to some kind of translation book trade show (I guess it is a thing) and hyped my book up and two publishers in China wanted to translate it and publish it there. They each submitted bids to the publisher, who then shared the bids with me and let me choose between them (the differences were just how much money they were going to pay for the exclusive rights to translate and publish it into Chinese, which goes to both me and the publisher, the price of the book they intend to sell, and the number of copies they intend to print). The Chinese publisher I chose now has the rights to the translation and publication and they will hire a translator. (This will take some time. They have agreed to send me a copy of the translation a week or two before they finalize it, which I'll probably run past a colleague who speaks Chinese just to make sure they spelled my name right.)

So the issue is less "who can translate it" (I mean, you could translate it for your own personal use, I don't think anyone would know or care, and I am sure that would be fair use in the USA), but "who can publish the translation," which is something that is going to depend on who has the copyrights on the original book (one could imagine a publishing contract different from mine, to be sure, but there is going to be some contract out there that describes who has the rights to translation). The author of the book may not be the person who owns the rights to translation publishing, depending on the contract they signed with the initial publisher.

Note that the above is basically identical to other kinds of "transformative" copyright processes, like creating an audiobook (which also happened for my book) — the publisher made some kind of deal with an audiobook company that then produced it, and I got to have the final "OK" on the choice of the company but that's about it.

Professional translator here. Being bilingual does not give someone the qualifications to be a translator, just like knowing how to write does not give someone the qualifications to be a writer. Some translators are self-taught, it is true, but bachelor's, master's and doctorate degrees in translation exist for a reason.

A translator can always contact a writer or their publisher and propose to translate their book. But it is often necessary to already have a deal with the target language publisher before even starting the project, as a translation without a publisher will amount to little result. Usually, the target language publisher will buy the rights to the book and then hire the translator.

It is also important to remember that English is not a language that translates much. Around the world, English is usually the language being translated. I'm not saying it never happens, but it is much less likely that a German book will be translated into English than an English book will be translated into German.

I, who have involved with a few translation project of the academic history book (as well as a quasi popular history book) either as one of co-translators or as an assistant translator in a non-Anglophone country, also confirm what the other comments basically say: It's primarily a negotiation between the publishers that have much voice to say, and the agency company also often involves to mediate the exact legal terms, such as acquiring the official publishing right of the translation, in many cases, at least in my country.

While it is not entirely impossible for the aspiring translator to submit to the proposal of the translation project on their own initiative, it has got increasingly difficult to persuade the publisher especially for the academic history book in recent years since less and less copies have been sold and many humanities became nearly bankrupt. So, the initiative often comes from the publisher that will publish the translation, not from the aspiring translator.

In addition to the translation right itself, a competent editor in such a publisher are often required to be good at, in addition to the ordinary jobs for otherwise non-academic publishing:

• how to find the source of possible additional funding - academic history books are very expensive (even compared with the original) in almost every case, and now often be sold only in the very limited copies (hundreds) anymore in my country. In some cases, however, to find and to apply the fund falls into the aspiring translator's side.
• the language itself (nearly) as good as the translator(s) themselves in some cases, to conduct the additional comparative check between the original text and the drafted translation. Due to this aspect, some academic publishers (with their editors) often have a linguistic specialty of the translated book.

they [the publisher] will hire a translator (cited from /u/Aquamarinade and /u/restricteddata below)

As for the purely academic humanity book, to hire a professional translator (alone) is not so likely the option in my country due to the following factors:

• Academic books are often sold in the very limited numbers, so the reward will be extremely tough in many cases - sometimes with a few copies for free only, and additional copies with the discounted prices (Except for the popular history book, all of my past involvement of the translation project in fact ended in the deficit in purely money terms - since I had to give some additional copies of the translated books as reward of my gratitude to whom I had asked some inquiries like those on proper names in their specialty).
• Not only to translate the original text, the translation of many academic history or humanity book requires the comparison between the cited works and their possible preceding translations in the local languages (To give an example, to check the citation of Max Weber's Wirtschaft und Gesellschaft with the well-known translation in the local language) in the foot/end notes section. Without access to the good library, it is sometimes not easy to progress this part of the process, and the affiliation with some higher education institution like the university will be ideal. It is also the time-consuming task, and many non-academic publisher just hate the notes (since it takes much time of translating as well as space in the published book). Some inferior quality translations just omit this process - in some cases, they omit the entire notes section in the translation (and thus nearly useless for the academic history book)!
• The spelling of some not so familiar person/ place names might also be checked and "standardized" in accordance with the practice of local language. The translator sometimes ask their colleagues or friends to check this kind of trivial issue personally.

Due to this additional condition of the academic book's translation, it is not so easy to find the professional translator who is willingly to accept the offer.

University professors and lecturers, with better access to the library and some "assistant" co-translators of the graduate student in relevant field, have instead usually been preferred either as the translator or as the supervisor/chief-translator of the total translation project of such academic books. The translation had been counted also as an academic achievement in humanities in my country, however, now almost not any more (so it will also become difficult to find possible stocks of such translators even among the professors).

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They have agreed to send me a copy of the translation a week or two before they finalize it, which I'll probably run past a colleague who speaks Chinese just to make sure they spelled my name right.

If the main translator as well as the editor are competent and meticulous, the original author might get in contact with the following two additional "appendixes" for the translation in some cases:

• To check some possible simple facts or factual errors found in the original text like the year - the translator is sometimes asked to add local annotations for readers in the localized edition.
• In some cases, the translator also asks the original author for a short comment to their localized edition.

(Edited): fixes typos and some time tenses (sorry).