Assuming the join table follows good practice and has a unique compound key defined, i.e. a constraint to prevent duplicate rows, then something like the following simple query should do.
select conversation_id from conversations_users where user_id in (1, 2)
group by conversation_id having count(*) = 2
It's important to note that the number 2 at the end is the length of the list of user_ids. That obviously needs to change if the user_id list changes length. If you can't assume your join table doesn't contain duplicates, change "count(*)" to "count(distinct user_id)" at some possible cost in performance.
This query finds all conversations that include all the specified users even if the conversation also includes additional users.
If you want only conversations with exactly the specified set of users, one approach is to use a nested subquery in the where clause as below. Note, first and last lines are the same as the original query, only the middle two lines are new.
select conversation_id from conversations_users where user_id in (1, 2)
and conversation_id not in
(select conversation_id from conversations_users where user_id not in (1,2))
group by conversation_id having count(*) = 2
Equivalently, you can use a set difference operator if your database supports it. Here is an example in Oracle syntax. (For Postgres or DB2, change the keyword "minus" to "except.)
select conversation_id from conversations_users where user_id in (1, 2)
group by conversation_id having count(*) = 2
minus
select conversation_id from conversations_users where user_id not in (1,2)
A good query optimizer should treat the last two variations identically, but check with your particular database to be sure. For example, the Oracle 11GR2 query plan sorts the two sets of conversation ids before applying the minus operator, but skips the sort step for the last query. So either query plan could be faster depending on multiple factors such as the number of rows, cores, cache, indices etc.