If an Eastern Rite Catholic marries in the Western rite, does he/she need to formally change rites?

TL;DR: No one has to translate to another Church sui iuris, but in some cases one spouse can do so (without complex procedure).

(Everything under the reservation, that the Rome decreed nothing special for special circumstances your case belongs to. But I do not think there is such a decree/law/rescript/...)

The ascription to a Church sui iuris (Eastern Catholic Churches and Latin Church) is ruled in cann. 29ff. CCEO and cann. 111f. CIC. The relevant norms for us are:

CCEO:

Can. 33 - A wife is at liberty to transfer to the Church of the husband at the celebration of or during the marriage; when the marriage has ended, she can freely return to the original Church sui iuris.

Can. 36 - The transfer to another Church sui iuris takes effect at the moment a declaration is made before the local hierarch or the proper pastor of the same Church or a priest delegated by either of them and two witnesses, unless the rescript of the Apostolic See provides otherwise.

Can. 37 - Every enrollment in a certain Church sui iuris or transfer to another Church sui iuris should be recorded in the baptismal register of the parish where the baptism was celebrated, even, as the case may be, in a Latin parish; if this cannot be done, it is to be kept by the proper pastor in another document in the archive of the parish of the Church sui iuris of enrollment.

CIC:

Can. 112 - § 1. After the reception of baptism, the following are enrolled in another Church sui iuris:

1° a person who has obtained permission from the Apostolic See;

2° a spouse who, at the time of or during marriage, has declared that he or she is transferring to the Church sui iuris of the other spouse; when the marriage has ended, however, the person can freely return to the Latin Church;

3° before the completion of the fourteenth year of age, the children of those mentioned in nn. 1 and 2 as well as, in a mixed marriage, the children of the Catholic party who has legitimately transferred to another Church sui iuris; on completion of their fourteenth year, however, they can return to the Latin Church.

§ 2. The practice, however prolonged, of receiving the sacraments according to the rite of another Church sui iuris does not entail enrollment in that Church

§ 3. All those transfers to another Church sui iuris have force from the moment of the declaration of the fact before the local Ordinary of the Church or the proper pastor or priest by delegation and two witnesses, unless a rescript of the Apostolic See provides otherwise; and [this is to be] noted in the baptismal register.

So we see: No one is forced to transfer to another Church sui iuris at marriage. We have to differntiate three cases [Pree/Berkmann, n. 5]: 1. A man of Eastern Church sui iuris A marries a women of Eastern Church sui iuris B: The woman can decide to transfer to Chruch A, can. 33 CCEO. The man cannot transfer. 2. A man of Eastern Church sui iuris A marries a women of the Latin Church: The women can decide to transfer to Church A, can. 33 CCEO. The man cannot transfer. 3. A man of the Latin Church marries a women of Eastern Church sui iuris B: The women can decide to transfer to the Latin Church, can. 33 CCEO. The man can decide to transfer to the Church B, can. 112 § 1 n. 1 CIC.

In all cases it is irrelevant, in which rite the wedding is celebrated. (It has to be the correct priest for the rite and the priest has to be responsible for the wedding, i.e. normally: one of the spouses is in his Church sui iuris and the wedding is in his area.)

So the information you got is not fully correct: The Latin women can, but not has to, translate to the Church sui iuris of her husband. He cannot translate to the Latin Church, even if they marry in Latin Rite. (BTW: There is no one and only "Eastern Rite".)

If one of the spouses wants to translate to another Church sui iuris, they have to make a formal declaration of this before his/her new bishop (patriarch etc.) or pastor (can. 36 CCEO, can. 112 § 3 CIC). The transfer has to be recorded in the baptismal register (can. 37 CCEO, can. 112 § 3 CIC). This can be done at the earliest during the wedding, but later it is possible too.

To show that it is very important for the Church, that a transfer to another Church sui iuris is voluntary and an extraordinary act, some more norms:

Can. 31 CCEO - No one can presume in any way to induce the Christian faithful to transfer to another Church sui iuris.

Can. 1465 - One who, belonging to any Church sui iuris, including the Latin Church, and exercising an office, a ministry or another function in the Church, has presumed to induce any member of the Christian faithful whatever to transfer to another Church sui iuris, contrary to can. 31 is to be punished with an appropriate penalty.

Can. 32 - § 1. No one can validly transfer to another Church sui iuris without the consent of the Apostolic See.

§ 2. In the case of Christian faithful of an eparchy of a certain Church sui iuris who petition to transfer to another Church sui iuris which has its own eparchy in the same territory, this consent of the Apostolic See is presumed, provided that the eparchial bishops of both eparchies consent to the transfer in writing.

The consent of the Apostolic see or the bishops (can. 32 CCEO) is not needed in case of marriage (can. 33 CCEO).

Can. 38 - Christian faithful of Eastern Churches even if committed to the care of a hierarch or pastor of another Church sui iuris, nevertheless remain enrolled in their own Church.

[Pree/Berkmann]: Helmuth Pree/Burkhard J. Berkmann: Commentary on can. 112 CIC. in: Klaus Lüdicke (Ed.): Münsterischer Kommentar zum Codex Iuris Canonici. unter besonderer Berücksichtigung der Rechtslage in Deutschland, Österreich und der Schweiz. Ludgerus Verlag. Essen, Germany (state: April 2017)