Hermione would likely try to outwit them. She has a sharp mind. She might use her knowledge of magic, both dark and light, to find ways to counter the Death Eaters. For example, she could use spells that they wouldn't expect or find loopholes in their dark magic.
She could be a target. The Death Eater Inner Circle is known for their dark magic and evil deeds. Hermione, being a smart and powerful witch, might be seen as a threat to their plans. So they could try to capture or eliminate her.
Over time, if she couldn't immediately escape, Hermione would try to influence the Death Eater. She would use her knowledge and strong moral compass to show him the error of his ways. She might start by having debates with him about the Dark Arts and the ideology of the Death Eaters, hoping to make him see the light.
She could also try to reason with the Death Eater. Hermione is known for her intelligence and ability to think on her feet. She might try to appeal to any sense of humanity the Death Eater might have, reminding him of the consequences of following Voldemort's evil path.
She might react with caution at first. Given that Death Eaters are known for their cruelty, Hermione would be smart to be on her guard. She would likely start by gathering information about the escaped Death Eater, like his known abilities and any possible weaknesses.
Some in the wizarding world might see it as a chance for redemption for the Death Eater. If Hermione can reform him, it could be seen as a positive thing. But most would still be very cautious and watchful, fearing that it's all a ruse by the Death Eaters to cause more chaos.
She might be very cautious at first. Hermione is a smart and brave character. She would probably try to gather as much information as she could about the Death Eater without being detected. Maybe she'd use a spell to keep herself hidden while observing the Death Eater's actions.
She could be angry. Hermione is used to being in control of situations through her knowledge and planning. Having a Death Eater surprise her disrupts that. So she might feel angry at the intrusion and at the threat this Death Eater poses to her safety and the safety of those she cares about. This anger could fuel her actions in the encounter, making her more aggressive in her defense.
In some fanfictions, Hermione as a Death Eater is a very interesting and often controversial portrayal. Sometimes, it might show her being forced into the role due to some dark magic or blackmail. This completely flips her good - girl image from the original series and allows for a very different exploration of her character.
It might be portrayed as a tense encounter. Hermione would be on her guard, ready to use her magic if necessary. The Death Eater, on the other hand, could be sneering and full of dark arrogance.
Hermione would use her vast knowledge of magic. For example, she could use concealment charms to hide the Death Eater from other Death Eaters who might be looking for him. Also, she might use polyjuice potion to disguise him if they need to move around in public.