No.
Any individual can gift up to $12,000 per year to any other individual. Any amount over that is subject to gift tax. The giver would be required to file a gift tax return, and pay taxes once the $1,000,000 exemption amount is used up. Gifted amounts also reduce the estate tax exemption amount.
Your mother would have to file a gift tax return for the value of her half share of the vacation home that she gives you (less $12,000). Whether or not she actually has to pay gift tax (up to 45%, if I remember correctly) depends on whether or not her $1,000,000 exemption amount has been used up.
If she sells her share for $1, it's the same thing. The difference between Fair Market Value (FMV) of her vacation home share and whatever you pay for it (i.e., $1) is considered a gift which would have to be reported on a gift tax return.
As a gift (which it is even if you pay $1), you would also receive your mother's cost basis, thus affecting the capital gains tax you would owe if you sell your share. Unlike inherited assets, there is no "step up basis" for gifted assets.
If you buy your mother's share for FMV, then no gift tax is involved, but your mother would owe capital gains tax on the difference between her cost basis and the sale price.