Between 4-7 months, babies begin to understand object permanence, meaning they know mom and dad are the most constant thing in their life. If you're not in the same room, they'll recognize that. You may be in the kitchen or next room, but they don't have a concept of time, so all they know, you're gone forever, so if you leave, they cry. This is normal. Actual separation anxiety usually starts between 8 months and their 1st birthday. This is when they may have a hard time being apart from mom or dad. It's a normal part of development and you can take comfort in the fact that they have a healthy bond with you. But it's good for you to look for opportunities to leave them in the care of someone else. Even if this isn't something you do on a regular basis, it's good for them to get used to being without you, and it's good for you to get out and do other things.
If you anticipate having to leave your baby with someone on a regular basis, try to do it before your baby is 8 months. But rest assured that if you've found a trusted caregiver, you can leave your baby in their care. Your baby is only going to cry for a few minutes (in most cases) after you've left. You're the one who may feel guilt and dwell on it more than the child after you've left. It's important to develop a ritual when you're leaving your baby with someone - something calming and comforting. Have confidence in your child when you talk to them. Use positive expressions and a happy voice. Tell them that you'll be coming back. They'll learn to trust you as you do that time and time again. After you've left, distraction is key, so if there are comforting items you can leave with the caregiver that they can use as distractions, that may help the transition go more smoothly. Try to not leave your baby with someone when they're really tired or hungry, because all their emotions are heightened at those times.
Look for opportunities between 4-7 months and beyond to let your baby learn that you're going to come back. You can do this at home. Leave them (after they've been fed, changed and loved) in a playpen or a safe, contained area while you go in another room and fold laundry or do dishes, but can keep an eye on them. They may cry for a couple minutes when they can't see you, but it's better if you can see them so you know they're okay. Let them cry for a couple minutes, learn how to self-soothe, distract themselves and move on from the fact that they're anxious about you not being there. They'll learn that when you leave the room, you come back. They learn through experience.
If you feel like you have a newborn that's struggling with separation anxiety, it isn't due to traditional separation anxiety. It's normal for a baby to cry when you place them down, because they love you, they want you to hold them and it's comforting to them. It's not realistic to hold your baby all the time. There are some things that you have to get done. You have a couple options. One is feeding, changing, loving them and putting them in a playpen in the same room you're in while you get things done. If they cry and you know they're crying because they want you, then it's okay to let them cry while you finish things you need to attend to. You also have the option of using wraps. These are helpful. I used one and I was able to get things done while I wore him. He was close to my chest and happy to be with me while I vacuumed, did dishes and did things around the house. Just be sure that if your baby is falling asleep, that you put them down and don't have them sleep on your chest with the wrap around you, because doing that has been associated with harm to the baby.
Another good thing you can do with babies is peek-a-boo. You cover your face, you know you're there, but they don't, so it's a surprise every time they see your face, but they learn that you come back. That's part of teaching them. You can play it when they're little and continue as they get older. When they get older and you're leaving them places, for example, my son would go to daycare and I'd always tell him (once he was old enough to understand) I was going to come back after playing outside. He knew I'd pick him up after he played outside, which was around 4:30, but he doesn't understand 4:30, so you need to talk about it in terms the child will understand. And they'll hold you to it, so keep your promises. That's part of helping them learn to trust you'll come back when you say.
If you have specific questions or concerns, or you've noticed your child is having nightmares or meltdowns no matter what you're doing, or they seem disturbed about being left with certain people, then you should investigate. Under some circumstances, like if they're having nightmares or you're noticing behavioral changes, talk with your pediatrician. They can ask you questions and (based on their knowledge of your circumstances) give you tailored information and advice.