Of course, there will be two separate DSNs, one for user1, the other for
user2.  If you use linked tables (I rarely do anymore, preferring pass
through queries or PTQs), then it should be a simple matter of having
the correct tables links.

What I prefer to do is use PTQs is to run updates via VBA, something
like this (sounds like you have some experience with VBA, if not let me
know).  This is longish, but go through it slowly if you're relatively
new to this:

'code start AIR CODE ALERT! ****************************

Function fRunPtq(strS as string, strConnect as string) as Boolean

'strS is the ORACLE action query (update, delete, insert)
'Expressed in Oraclese, not Jet.
'
'strConnect is a connect string constant

  dim dbs as DAO.database
  dim qdf1 as DAO.querydef

  set dbs = access.currentdb

  'Make a temporary querydef, ie, name = ""

  set qdf = dbs.createquerydef("")

  with qdf

    .connect = strConnect

    .SQL = strS

    .returnsrecords = false

    .execute dbfailonerror

    .close

  end with

Exit_Proc:

  Set qdf = nothing

  dbs.close

  Set dbs = nothing

  Exit Function

Err_Proc:

'no error handling included here for this post, but error procs can be
included to return a false for this function of the query does not execute.

End Function

'Code end*******************************

In a standard module, I would have the connect strings as public
constants as follows (watch wrap) which is copied from one of my own
apps and passwords, user names changed(!).  Note each string is for
either of the two different users, user1 and user2:

'COde start *******************************

Option Compare Database
Option Explicit

'Connect Strings

'For USER1 Oracle user

Public Const cUser1Connect =
"ODBC;DSN=OOGA;UID=USER1;PWD=whatever;DBQ=tma;DBA=W;APA=T;PFC=1;TLO=0;DATABASE="

'For USER2 Oracle user

Public Const cUser2Connect = "ODBC;DSN=Ooga2
XXX;UID=USER2;PWD=whatever;DBQ=TMA;DBA=W;APA=T;FEN=T;QTO=T;FRC=10;FDL=10;LOB=T;RST=T;FRL=F;MTS=F;CSR=F;PFC=10;TLO=0;"

'Code End*******************************

To get the proper DSNs, all you need do is create a pass through query
on the query design screen and set the connect string property there.
Then copy and paste as above.

Now, as an example, to run two separate update statements (queries) in
the two separate users, I generally write the Oracle SQL code as part of
the VBA function.  The following is air code.

'code start AIR CODE ALERT! ****************************

Sub sExample()

'No error handling shown here

  dim strSql as string 'for the Oracle SQL

  'The first will be an update against a table in USer1

  strsql = "UPDATE MyOracleTableA Set MyColumn = 'A different value'

  'Now call the above function and run this sql.  Note the use
  'of cUser1Connect:

  fRunPtq strSql, cUser1Connect

  'Now, an update against a table in USer2

  strsql = "UPDATE SomeOtherOracleTable Set AnotherColumn = 'A
different    value'

  'Again, the above function runs this different statement.
  'Note the use of cUser2Connect this time:

  fRunPtq strSql, cUser2Connect

  'And so on.

end Sub

'Code End*******************************

Note that with different users, you may need to express a user name in
front of the table name.

Hopefully you are able to follow this.  If you have difficulty, go
slowly, I've used a considerable amount of VBA.

Let me know if I can clarify or help further.