There are at least three kinds of testing involved with a localization project: linguistic, cosmetic, and functional.
Linguistic testing is the most familiar to translators, as we test the quality of the translation all the time. This might also be done by a third party like an in-country reviewer. We are checking that the correct terminology is used and that the translation is understandable and consistent with the feel of the source language. Sometimes the translation is perfect, but a string appears in a context that you didn’t expect and where it doesn’t make sense. If possible, test your changes by running the software. Sometimes you won’t have access to the development environment and this won’t be possible. In this case, ask to be sent screen shots from the integrated product for you to review.
Cosmetic testing is checking that the look, feel, and flow of the software hasn’t been changed by the linguistic changes. Primarily, we’re checking that expansion hasn’t caused text to be truncated or hidden, or broken the layout of the page. The customer should test this as well, but the first responsibility here is with the L10n engineer. Again, you may need to ask for screen shots to test this properly.
Functional testing is usually done by the product team. This usually doesn’t involve L10n engineers, but i18n engineers may be involved, and i18n engineers should always do functional tests of their own changes. Any time we make changes to software code there is a risk that we will introduce a defect. Defect-free code is the exception, not the rule. We can create spectacular failures just by deleting the wrong semi-colon or adding a quotation mark where it doesn’t belong. So even when we are making minor linguistic changes, the changes need to be integrated and tested to make sure everything still works.
An exception where you might do functional testing on your own changes is if you are provided a test framework. This is a separate computer program that will generally run a suite of tests that validate the basic functioning of the software. If you are given such a framework you should confirm that it runs successfully before turning in the work.