The gemara says that when one finds something in a public area where there are rov goyim, even if a yisroel would give a siman he doesn't have to return it. But, he should return it lifnim m'shuras hadin. Just as the case where the father of Shmuel found donkeys in the midbar and returned it after 12 months do fulfill the lifnim m'shuras hadin. The gemara implies that after 12 months there is no mitzvah to return a lost object. After 12 months of what? Tosafos explains that after 12 months the finder shouldn't continue to feed the animals because the animals will eat more than their value (which the owner will have to compensate for). Therefore, for 12 months the finder should continue to feed them (and eventually charge the owner for the expense), but after 12 months he should sell them and return their value to the owner. The father of Shmuel went lifnim mishuras hadin by returning the actual donkeys even after 12 months have passed. Tosafos seems to imply that the father of Shmuel accepted upon himself the loss of feeding the animals at his own expense after the 12 months, because if he continued to charge for the food, he would be causing the looser more harm than good by keeping the actual animals. The problem is that Tosafos writes in the very next Tosafos (d.h. lifnim) that lifnim mishuras hadin doesn't demand that one suffer a loss of money - so what did the father of Shmuel do!
Rashi learns this gemara completely differently. Any object that has been lost for more than 12 months has an automatic yiush. Once the object has been lost for 12 months, the finder would be entitled to keep it, yet the father of shmuel went lifnim mishuras hadin and returned it even after 12 months passed. Rashi goes lishitaso in Brachos 58b where the gemara says a dead person is forgotten after 12 months, quoting a pasuk comparing a dead person to a lost vessel. Rashi explains:
ככלי אובד - וסתם כלי לאחר שנים עשר חדש משתכח מן הלב דיאוש בעלים לאחר שנים עשר חדש בפרק אלו מציאות דף כח ע"א מי שמצא כלי או שום מציאה חייב להכריז שלש רגלים, ואם נמצא אחר הסוכות צריך להמתין ולהכריז בפסח ובעצרת ובחג, דהיינו שנים עשר חדש ושוב א"צ להכריז
Rashi assumes that the gemara which exempts one from calling out after 3 regalim pass really means 12 months and is based on the assumption of yiush (rather than just an exemption from going through the trouble of calling it out forever). Therefore, Rashi in our sugya holds that after 12 months of it being lost one would be allowed to keep it (unless he picked it up within the 12 months and becomes a shomer on it - באיסורא אתא לידיה). The comparison of Rashi between "forgetting" someone who died, and yiush sheds some light in the understanding of what yiush is. Rashi seems to hold that yiush doesn't require a formal statement of ווי לה לחסרון כיס, rather it is just a passive position where one "forgets" the lost object no longer considering himself an owner over it. This approach is very meduyak in rashi 21b (d.h. mei'hashta) where rashi explains that according to rava we view it as if yiush is taking place now, prior to him realizing that it is lost - שהרי נפל, וכשיודע שוב אין דעתו עליו. What does Rashi mean by "וכשיודע שוב אין דעתו עליו"? He means to say that had yiush required one making an active declaration then rava would admit that it couldn't go into effect prior to him even knowing that it is lost. But yiush doesn't require an active declaration, rather it is defined as "not having da'as on it", which is a very passive stance of no longer considering oneself an owner over the object. That is why rava would allow the yiush to take effect even before the person knows that it is lost - because had he been privy to that information he wouldn't consider himself an owner over the object, which is really the definition of yi'ush (see dibros moshe he'ara #7).