I think it is unfair to say that one who has no degree in the field of interest cannot become, by training and experience, a technical expert in that area, to deliver quality translation work.
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Translators are, or become, by the nature of our work, not just linguists, but technicians, researchers, scientists, writers, consultants and scholars as well - with or without formal training.
And if clients want a super qualified translator, they should also know that it does not come cheap. I know this has become a huge issue, with rates dropping, professionals from all over the world - different standards and costs of life - taking their place in the sun.
College and continuing education, even with the Internet and all, are still, expensive, and we should be paid accordingly for our work, for our investment in knowledge so we can serve the world, don't you think?
And if you want to stand out in this increasingly competitive industry, it is essential, take it from me, that you leave your comfort zone, and look to master something other than your black-belt kung-fu right now!
Believe me, even if you've just started in this trade, you should start working to become a "navy seal" in a high-demand area as soon as you can! Even if it is an area that reminds you of those veggies your mom used to force you to eat when you were a child!
Don't just give up!
If you consider yourself a specialist in a field, but you don’t have that college degree or technical diploma the client requires, make sure you have a portfolio with sample translations, published works, and clients who are willing to recommend you.
Still, sometimes it is just better to go back to college if you can. Take medical or healthcare related courses, participate in extension or vocational programs, go into new areas, or refresh what you know!
Stay at the top of your game, adapt to the market's needs.
It is never too late to go back to school, you know!
Take the first step.
If you want to start in the medical translation industry, I recommend you introduce yourself to the terminology by taking a free course on the subject, such as the one offered by Des Moines University https://www.dmu.edu/medterms/.
There are also webinars and online courses on ProZ.com, designed and instructed by experienced professionals who are willing to share their secrets to success.
You can also join the medical translation chapters of your translation association, or even become a student member of the American Medical Writers Association (if English is your target language), and take their "Essential Skills Certificate Program" for starting medical writers.