The answer to your question is that it implies nothing and neither.
You cannot know for sure without getting explicit confirmation. People will twist things to their own advantage each and every time. The only prudent course is to assume the worst case.
If I expect something to be done by next week, then when next week rolls around, it had best be done already. That means by the start of the week. However, the person whom I’ve directed to do it will naturally want to dillydally and delay until the end of the week. No joy will be found here on either side.
Don’t guess: get it in writing, and get it in UTC. For example, next year’s vernal equinox occurs on March 20th of 2013 at 11:02 UTC. That is an exact time. Find out if that is when they expect it done. If they say no, make them tell you when. Exactly when.