Many of the other languages that do have eight distinct words for the directions are related to Estonian: Finnish, Ingrian, Karelian, Livonian, Ludic, Veps, and Votic. In addition to the Finnic languages, Breton and Sanskrit also have eight unrelated words for the eight directions. (Interestingly, the Breton word for "south", kreisteiz, also means "noon," just as the Estonian word for south lõuna does.)
Indonesian and Malaysian have a specific word for southeast, tenggara, but use the words for east (timur) and west (barat) in the words for southwest, northwest, and northeast (barat daya, barat laut, and timur laut).
Most other languages of the world simply combine the words for the four cardinal directions to create the other four directions between them, like English does.
English: northeast, southeast, southwest, northwest.
Spanish: noreste, sureste, suroeste, noroeste.
French: nord-est, sud-est, sud-ouest, nord-ouest.
German: Nordosten, Südosten, Südwesten, Nordwesten.
Swedish: nordost, sydost, sydväst, nordväst.
Russian: северо-восток, юго-восток, юго-запад, северо-запад.
Chinese: 东北,东南,西南,西北.
Estonian: kirre, kagu, edel, loe.
Finnish: koillinen, kaakko, lounas, luode.
Image source (pildi allikas): Jimpaz via Wikimedia Commons.
Several of these words (notably kirre, lääs, and loe) have significant changes in other case forms, so I will list the relevant forms below.
North:
nimetav ainsus (nominative singular): põhi
omastav ainsus (genitive singular): põhja
osastav ainsus (partitive singular): põhja
osastav mitmus (partitive plural): põhju / põhjasid
Northeast:
nimetav ainsus (nominative singular): kirre
omastav ainsus (genitive singular): kirde
osastav ainsus (partitive singular): kirret
osastav mitmus (partitive plural): kirdeid
East:
nimetav ainsus (nominative singular): ida
omastav ainsus (genitive singular): ida
osastav ainsus (partitive singular): ida
osastav mitmus (partitive plural): idasid
Southeast:
nimetav ainsus (nominative singular): kagu
omastav ainsus (genitive singular): kagu
osastav ainsus (partitive singular): kagu
osastav mitmus (partitive plural): kagusid
South:
nimetav ainsus (nominative singular): lõuna
omastav ainsus (genitive singular): lõuna
osastav ainsus (partitive singular): lõunat
osastav mitmus (partitive plural): lõunaid
Southwest:
nimetav ainsus (nominative singular): edel
omastav ainsus (genitive singular): edela
osastav ainsus (partitive singular): edelat
osastav mitmus (partitive plural): edelaid
West:
nimetav ainsus (nominative singular): lääs
omastav ainsus (genitive singular): lääne
osastav ainsus (partitive singular): läänt
osastav mitmus (partitive plural): lääsi
Northwest:
nimetav ainsus (nominative singular): loe
omastav ainsus (genitive singular): loode
osastav ainsus (partitive singular): loet
osastav mitmus (partitive plural): loodeid