Can you teach 2 dogs different behaviors?

I’m getting an 8-week old puppy in April. I already spend quite a lot of time with my neighbor’s chihuahua. She’s around 3, and is a very sweet dog, but she has some terrible manners. She is allowed on the bed and furniture due to her size. She begs for food by pawing at your hand or sitting practically on top of your plate, she licks non-stop, and will poop in the house. Her owner thinks it’s funny when she chases people and nips at their heels on our beach. Again, sweet dog, but no attempt to train her by her owner.

My pup will end up being 140 pounds, so I need to set his boundaries right away. He will not be allowed on furniture or in bed. I will not tolerate begging, licking, or pawing. I intend to train him in Search and Rescue, so obedience and manners will be important.

Can you train two dogs different behaviors? Will it affect him as far as being an alpha male if this tiny chihuahua doesn’t have as many rules as he does?

Not tending to my neighbor’s dog is not an option. She’s in her late 60s and has a hard time getting the dog outside for exercise and doesn’t really care about the things I mentioned above. Some of them aren’t offensive in a 10-pound dog, but most certainly would be in a 140-pound search dog. Plus, it isn’t my business how she trains her dog.

Thoughts? Ideas?