You cannot customise what product attributes get copied over when syncing a product over. However, after the copy, you are able to customise/change any product attribute you'd like (so long as you haven't enabled ongoing syncing via custom sync module) except for the SKU.

Through our Product Settings add on, you can also enable the 'Hide synced product by default' setting so that you can modify the product before it is published in your destination store. The product will be set as unpublished upon its copy across to the destination store.