you can always export / import the objects owned by u.
If You are looking into a full database export or export other users data, You need dba rights.

You can create a database link in the target.
and copy/extract the tables 
something like ( in the target)

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create database linke sourcedb_tnsname
connect to scott identified by tiger
using 'sourcedb_tnsname';
-- now create a full table or extract certain columns
-- or use where conditions to create your new tables
-- from the source tables over the databaselink

create table local_emp select * from emp@sourcedb_tnsname;