INDEX Creation Advice Oracle
Whether you use a local index or a global index will have (practically) no impact on the amount of space required for the index.  If you are running out of space to create the index, you'll need to allocate additional disk space to the tablespace (or tablespace(s) if you create a local index where the index partitions are stored in different tablespaces).  If you are building an index that involves the column you are partitioning on, you would almost always want to create a local index-- there is no benefit to creating a global index in this case.

Assuming that the dob column stores dates of birth and that the time component of that column is always set to midnight, you can probably reduce the space required for the index by making the dob column the leading column of the index and by using index compression.  In your case, you probably want to specify COMPRESS 1 since you would only want to compress the dob column, not the person_id column assuming that person_id has many fewer repeated values.