After Leah gives birth to 4 boys, Rachel, following a confrontation with Yaakov, gives over her maidservant to Yaakov so that he may produce children with her. She declares with utmost certainty, (30:3) "ve'ibaneh gam anochi mimenah," and I, too, will be built up through her. This statement is in noticeable contrast to Sarah's statement when she gives Hagar to Avraham. There, she states (16:2) "ulai ibaneh mimenah," perhaps I will be built up through her.
There's a simple explanation for the different approaches taken by the imahos. Although the Torah states that Sarah was barren, without children, Avraham was equally childless. The exact cause of their childlessness was seemingly unknown. Had Avraham been the infertile one, giving him Hagar would not have helped.
When Manoach and his wife were childless before the birth of Shimshon (Shofetim 13), the midrash (Vayikra Rabba 9) recounts that there was a conflict between them as two who was responsible. That is why the angel appeared to her to tell her that she was the infertile one, but that they would soon have a child. We see from Sarah's handling of her situation that no such conflict existed between Avraham and Sarah. Sarah was perfectly ready to accept that she was the barren one and have Avraham reproduce through Hagar, albeit with that slight hint of uncertainty.
The situation with Yaakov and Rachel, of course, was completely different. Yaakov had already fathered four children with Leah. Rachel knew that she was barren and Yaakov was not. She had no reason to worry that Yaakov's union with Bilhah would not produce children and therefore, was certain that she would be built up through her.