On my local computer, where is the data stored for a search on a dynamodb database I am using RazorSQL to access a dynamodb database in a "SQL"-like way. 
 One of the things RazorSQL lets you do is the sql command select * from table where ... on all columns. This is significant better than Amazon as Amazon only lets you do the filter on the hash and primary_key columns.

My question is whether the search is performed on my local computer or on the Amazon servers.

This matters because if my database has 100 lines and my search returns 3, I should only get charged for 3 rows of data retrieval if the search happens on the Amazon end, but if the search happens on my end, thus necessitating transfer of the datatable to my computer, I would get charged for 100 retrievals.
The search is made on the server host where the database service is installed. And you will only get the result set sent to you. 
  It would make no sense your client to make this, as your client is just an interface that allows your to interact with your database.
 This is available for all databases.