Postgres full text search with multiple columns, why concat in index and not at runtime?
No you don't need separate indexes. Use the weights feature. They are just a label your can query against. You can have up to four labels to query against (A-D). 

--search any "field" for quick:
select 'quick:1A brown:2B quick:3C'::tsvector @@ 'quick'::tsquery; --true

--search B "field" for quick:
select 'quick:1A brown:2B quick:3C'::tsvector @@ 'quick:B'::tsquery; --false

--search B or C "fields" for quick:
select 'quick:1A brown:2B quick:3C'::tsvector @@ 'quick:BC'::tsquery; --true


You might want to concatenate tsvectors, so that you can separately apply weights to them and then put them together:

select
  setweight( name_column::tsvector, 'A') || setweight( phone_column::tsvector, 'B');