I will try to answer your question as clearly as I can.
Access doesn't do anything differently at all for local vs. remote databases as long as both are .MDB files. However, Windows will do something differently. If the database is not an Access file but is instead something in MySQL or ORACLE or one of the others for which Access has adequate ODBC libraries, then and only then will Access do something different.
When Access handles a query, whether explicit or implied (as for forms and reports), it ALWAYS reads the tables involved in the query one record at a time in prime key order to form a temporary recordset as the result of the query. The temporary recordset is embodied as a series of pointers to table records to be presented in the order implied by the query.
The only difference for Access databases is whether the disk driver hits a physical disk or the network driver hits a remote disk when you issue the read or write request. That's the ONLY difference for Access local and remote databases. Windows makes the operation transparent to Access regarding disk location. (Which it should.)
When ODBC is involved, however, Access does not read or write records right away. Instead, it constructs the SQL needed for the explicit or implied query and sends that through the ODBC link. The database manager program at the other end of the link does all the work of forming the resulting recordset. Then it presents that result set one record at a time through the ODBC link under terms already negotiated when you opened the recordset in the first place. The difference is, of course, that all disk operations were performed by the program on the opposite end of the ODBC link from Access. Presumably, all database files are local to that program, so no network delays are incurred in forming the recordset. No other network users have to bear the implied burden of scanning your gigabyte database to form the result set.
What surprises me the most is that your gigabyte database hasn't already ticked off your users. Do you have a gigabit Ethernet backbone there?
In any case, performance in the all-Access case is best improved by assuring that you have keys available for all large queries. Access will use the keys to help it form the required recordsets when they are available AND APPLICABLE. So tailor some keys for your biggest, most popular queries/reports.