The two cases are different.
At the first case both columns needs to be searched for the same value. If you have a two column index (u1,u2) then it may be used at the column u1 as it cannot be used at column u2. If you have two indexes separate for u1 and u2 probably both of them will be used. The choice comes from statistics based on how many rows are expected to be returned. If returned rows expected few an index seek will be selected, if the appropriate index is available. If the number is high a scan is preferable, either table or index.
At the second case again both columns need to be checked again, but within each search there are two sub-searches where the second sub-search will be upon the results of the first one, due to the AND condition. Here it matters more and two indexes u1 and u2 will help as any field chosen to be searched first will have an index. The choice to use an index is like i describe above.
In either case however every OR will force 1 more search or set of searches. So the proposed solution of breaking using union does not hinder more as the table will be searched x times no matter 1 select with OR(s) or x selects with union and no matter index selection and type of search (seek or scan). As a result, since each select at the union get its own execution plan part, it is more likely that (single column) indexes will be used and finally get all row result sets from all parts around the OR(s). If you do not want to copy a large select statement to many unions you may get the primary key values and then select those or use a view to be sure the majority of the statement is in one place.
Finally, if you exclude the union option, there is a way to trick the optimizer to use a single index. Create a double index u1,u2 (or u2,u1 - whatever column has higher cardinality goes first) and modify your statement so all OR parts use all columns:
... WHERE (user_1 = '$user_id' OR user_2 = '$user_id') ...
will be converted to:
... WHERE ((user_1 = '$user_id' and user_2=user_2) OR (user_1=user_1 and user_2 = '$user_id')) ...
This way a double index (u1,u2) will be used at all times. Please not that this will work if columns are nullable and bypassing this with isnull or coalesce may cause index not to be selected. It will work with ansi nulls off however.