Married filing separately. Do I enter both persons' W2s, one after the other? Or do I later prepare a separate return for my spouse and enter his W2 then??

Since you are filing separate you enter your W2 on your own return. You prepare another return under their own account for your spouse & enter his information on his return.

Married Filing Joint vs. Married Filing Separately
https://ttlc.intuit.com/replies/3288477

Do not use your account to do your spouses return.

Online is only good for 1 return.  But you can use the same email address for 5 accounts.   How to start another return in the Online version
  https://ttlc.intuit.com/replies/3288574