Non-Chinese records of Zheng He's expeditions come primarily from Arab sources from Yemen. Southeast Asian history-keeping was still in a rather incipient stage^1 and as far as I know, things were similar in South India. Any Persian-language sources must have been compiled in the port of Hormuz that Zheng He visited and were presumably destroyed when the Safavid dynasty conquered the city in 1622 (maybe /u/Lion_and_Sun can verify this though?).

The most contemporary account of the expeditions is an untitled and anonymous annalistic account of the then-ruling Rasūlid dynasty of Yemen, compiled in the Islamic year 843 (1439-1440 CE). It reports the arrival of Chinese junks in 1419, 1423, and 1432. These approximately correspond to Zheng He's fifth, sixth, and seventh voyages. The 1419 arrival is described as:

Arrival of Dragon-ships [marākib al-zank] in the protected harbour city [of Aden] and with them the messengers of the ruler of China with brilliant gifts for his Majesty, the Sultan al-Malik al-Nāsir in the month of l’Hijja in the year 821 [January 1419]. His Majesty, the Sultan al-Malik al-Nāsiṛ’s in the Protected Dār al-Jund send the victorious al-Mahaṭṭa to accept the brilliant gifts of the ruler of China. It was a splendid present consisting of all manner of rarities [tuhaf], splendid Chinese silk cloth woven with gold [al-thiyāb al-kamkhāt al-mudhahhabah], top quality musk, storax [al-ʾūd al-ratḅ] and many kinds of chinaware vessels, the present being valued at twenty thousand Chinese mithqāl [93.6 kg gold]. It was accompanied by the Qādi Wajīh al-Dīn ʿAbd al-Rahman b. Jumay. And this was on 26 Muharram in the year 822 [March 19, 1419].

His majesty, the Sultan al-Malik al-Nāsir ordered that the Envoy of the ruler of China [rusul sāhib al-Sị̄n] returned with gifts of his own, including many rare, with frankincense wrapped coral trees, wild animals such as Oryx, wild ass, thousands of wild lion and tamed cheetahs. And they travelled in the company of Qādi Wajīh al-Dīn ʿAbd al-Rahman b. Jumay out of the sheltered harbour of Aden in the month of Safar of the year 822 [March 1419].

All three Chinese expeditions are said to have personally met the Yemeni sultan.

A later Yemeni historian, Ibn al-Daybaʿ (lived 1461-1537), recounts:

[The Chinese arrived at Aden in 1420 on] great vessels containing precious gifts, the value of which was twenty lacs [sic; lakhs] of gold... [Zheng He] had an audience with al-Malik al-Nāsir without kissing the ground in front of him, and said:

"Your Master the Lord of China [sāhib al-Sị̄n] greets you and counsels you to act justly to your subjects.”

And he [al-Malik al-Nāsir] said to him: "Welcome, and how nice of you to come!”

And he entertained him and settled him in the guesthouse. Al-Nāsir wrote a letter to the Lord of China:

"Yours it is to command and [my] country is your country."

He dispatched to him wild animals and splendid sultanic robes, an abundant quality, and ordered him to be escorted to the city of Aden.

Finally, a source from the Mamluk sultanate of Egypt briefly references what appears to be the ships of Zheng He's seventh voyage, which reached the Red Sea. Mamluk historian Ibn Taghrī Birdī's famous History of Egypt says:

Shawwāl 22 [21 June A.D. 1432]. A report came from Mecca the Honored that a number of junks had come from China to the seaports of India, and two of them had anchored in the port of Aden, but their goods, chinaware, silk, musk, and the like, were not disposed of there because of the disorder of the state of Yemen. The captains of these two junks wrote to the Sharīf Barakāt ibn Hasan ibn ʿAjlān, emir of Mecca, and to Saʾd al-Dīn Ibrāhīm ibn al-Marra, controller of Judda [Jeddah], asking permission to come to Judda. The two wrote to the Sultan about this, and made him eager for the large amount [of money] that would result if they came. The Sultan wrote to them to let them come to Judda, and to show them honor.

The best work on Arab sources on Zheng He has been done by Japanese historian Hikoichi Yajima in the 1970s. The Taghrī Birdī account hasn't been studied at all, really, except for two paragraphs by K. N. Chaudhuri, the historian who "accidentally came across" the episode.

^1 Traditional eighteenth-century Malay histories show only a vague awareness of Srivijaya.