Same problem here, on a fresh install of the OS and the browser
Trying to download the Brazilian Portuguese and English (United States) dictionaries.
Brave info
Version 1.10.97 Chromium: 83.0.4103.116 (Official Build) unknown (64-bit)
Package info
brave-bin 1:1.10.97-1
AUR Link
Error Message
Spell check dictionary download failed.
Please check with your network administrator to make sure that the firewall is not blocking downloads from Brave Software servers.
hosts file
127.0.0.1 localhost
::1 localhost
127.0.1.1 arch.localdomain arch
System information
Linux arch 5.7.5-arch1-1 #1 SMP PREEMPT Mon, 22 Jun 2020 08:10:02 +0000 x86_64 GNU/Linux