On a recent English > Korean project for an agency client, I gave the project manager the choice of translation-only by one professional translator or, for a bit more, translation and proofreading by separate linguists. Some clients work on a budget that doesn't allow the additional proofreading step at all, some jobs don't really warrant the extra effort, certain clients utilize a third-party proofreader after we deliver our translation, and quite a number (including all of my direct clients, since it's included in the base price) just have us handle both steps internally, where we deliver the original translation file and the intermediate file showing proofreader changes (for internal auditing purposes) and the final proofread files (for delivery to the end client).
Occasionally though I get feedback like, "Do you/your team feel comfortable enough to deliver this project without editing?" Another variation on this question is, "We don't have enough budget for a proofreader so please be sure the translator proofs his/her own work extra well."
This kind of client feedback puts me in a difficult situation. I don't want to admit that our translation service might be less than perfect at the base price (which is already not the lowest price in town), but it's also reasonable to expect and explain that adding a proofreader to the process will usually result in a better job. This is especially true for my teams that have been doing the translation-proofreading process for a a long time for me on thousands of projects.
In fact, when a client brings in a third-party proofreader or the end client uses an internal Korean speaker to proofread our work themselves, it isn't uncommon for the result to actually get worse because of the reviewer not understanding well what a proofreading process should encompass, and often results in new errors being introduced. This can also result in the sticky situation of whether or not to charge for also reviewing a reviewer's feedback.
At any rate, the way I answered the client on this question on the recent project is as follows:
It depends what the work is going to be used for. If just for the information, then trans-only is fine. If for publishing, well, multiple rounds of proofreading are pretty standard in the news business, and even for professional writers who aren't having to work in two languages. A second set of eyes is always a good idea, but of course, it does take time and cost money.