Proteins are made in neurons themselves.
We know that mRNAs (messenger RNAs) are important in protein synthesis. Secretory and membrane proteins mRNAs are translated along with the rough Endoplasmic Reticulum (ER) membrane, which is where proteins are made (rough ER is rough because it contains sites where protein synthesis occurs). So, most of the polypeptides (long chains of peptides that later become proteins) are transported across the rough ER membrane in a process call COTRANSLATIONAL TRANSFER.
Which makes sense, as a name, because things are being translated together (co-translation) and there is a transference of material across a cell.
When the entire polypeptide is transferred into the reticulum, a secretory protein is produced. Some proteins transported into the ER remain there; others are moved to other parts of the cell; still others are secreted into the extracellular space.