Sites are given full designs when they graduate from beta—though the beta/graduated terminology may be changing in the future, and the process could change as well.
In the meantime, you can discuss ideas for any site's design, logos, etc. on the associated Meta site. This is addressed in The 7 Essential Meta Questions of Every Beta:
- What should our logo and site design look like?
This one is pretty straightforward. Solicit contributions, throw out
ideas, post preliminary (or finished) designs, and be supportive and
respectful of other people’s ideas and creativity.
We have designers on staff who will actively help come up with site
designs but, if an idea stemming from the community stands out as
exceptional, we are happy to use it.
Some users feel that old blog post is not very useful any more:
In almost every beta, the site design question, for example, gets
asked and undergoes a fair bit of discussion even though a probable
graduation is too far into the future for this discussion to be
That said, the worst that will happen if you offer input is that it will be ignored. There's no harm in the discussion; be sure to use the search function first to see if there's an existing Q&A on that site to offer your design ideas or comments.