In Elastic Search, I think based off your information I would use 1 index. My understanding is users are only searching there own documents, and the documents seems to be relatively similar.
Performance - When searching you can use a Filtered Query to filter to only the documents matching the user. The user id filter is cache-able, and fast.
Scalable - In Elasticsearch, you control sharding and replication at index level. Elasticsearch can handle large numbers of indexes, I just think configuring appropriate shards and replications could be valuable for the entire index.
In a single index, you can still easy wipe away data (see delete by query) , and there should be little concern of seeing others data unless you write your queries wrong. A filtered query with that filters results to only those associated with a user id is very simple. Similar in complexity to searching a different index per user.
Your exact needs might fit a different approach better. Based what I have so far, I would do choose one index though.