How can I say of a person that he or she has an open or closed personality? Is there a standard adjective for that?
doing a search for: japanese adjectives describing personality
I quickly found this site: http://maggiesensei.com/2015/10/30/%E6%80%A7%E6%A0%BC-seikaku-how-to-describe-personality-in-japanese/
It has a large list of positive and negative personality adjectives, helpfully arranged in categories by similarities.
I'm not exactly sure what you mean by open/closed, but the site gives several words that could fit the idea of open:
★ 社交的 (な) = shakouteki (na) = outgoing
★ 明るい = akarui = cheerful, bright, happy
★ 元気 (な) = genki (na) = cheerful, energetic
★ 陽気 (な) = youki (na) = happy, vivacious, upbeat
★ 気さく(な) = kisaku = friendly
★ さっぱりしている = sappari shite iru = good sport
★ 活発 (な) = kappatsu (na) = active
★誠実 (な) = seijitsu (na) = sincere
★正直 (な) = shoujiki (na) = honest
★ 愛想がいい = aiso ga ii = friendly, affable
★ 話しやすい = hanashiyasui = to be easy to talk to
★ 打ち解けやすい = uchitokeyasui = to be easy to make friends with
★ 付き合いやすい = tsukiai yasui = to be easy to get along with
And as for "closed", you can take the last 4 in this list and use the opposite of the last word (ie instead of "ii" use "warui" and instead of "yasui" use "nikui") and the meaning would become the opposite.
Based on resources OP has provided in their comment, I don't think there is an single all-rounder word to translate the "open" or "close" personality.
The Quora answer,
To have an open personality is to be open to change. It’s someone who understands that people are imperfect and works with them. [...] Someone who is clear headed enough to see their flaws and admit them. [...] Someone who understands different people have different opinions and that each opinion is a reflection of their experience and observation.
- 頭が柔らかい — describes somebody has flexible thinking, though also could be understood as "quick-witted" ↔ (antonym) 頭が固い
- 開明的 — willing to accept new things, usually with a connotation to be "progressive" ↔ 頑固, 頑迷
11 a : characterized by ready accessibility and usually generous attitude: such as (1) : generous in giving (2) : willing to hear and consider or to accept and deal with
- 心が広い, おおらか — broad-minded and not being fussy ↔ 心が狭い, せせこましい
- 寛大, 寛容 — tolerant, sympathetic and generous ↔ 狭量
(3) : free from reserve or pretense
- 開放的, 開けっぴろげ — disclosive and less concealing; note that they are not compliment in some context ↔ 閉鎖的
The loanword オープン is generally associated with the last definition.