Starting my career as Data Scientist, is Software Engineering experience required?
1) I think that there's no need to question whether your background is adequate for a career in data science. CS degree IMHO is more than enough for data scientist from software engineering point of view. Having said that, theoretical knowledge is not very helpful without matching practical experience, so I would definitely try to enrich my experience through participating in additional school projects, internships or open source projects (maybe ones, focused on data science / machine learning / artificial intelligence).

2) I believe your concern about focusing on data science too early is unfounded, as long as you will be practicing software engineering either as a part of your data science job, or additionally in your spare time.

3) I find the following definition of a data scientist rather accurate and hope it will be helpful in your future career success:


  A data scientist is someone who is better at statistics than any
  software engineer and better at software engineering than any
  statistician.


P.S. Today's enormous number of various resources on data science topics is mind-blowing, but this open source curriculum for learning data science might fill some gaps between your BSc/MSc respective curricula and reality of the data science career (or, at least, provide some direction for further research and maybe answer some of your concerns): http://datasciencemasters.org, or on GitHub: https://github.com/datasciencemasters/go.