How do I wire half hot receptacle to 2 switches?
Three-Way Switches
You have standard toggle switches. You need three-way switches. A standard switches on/off. A three-way switch switches one common wire to alternate between two other traveler wires. Pretty much any standard toggle switch can be replaced with a 3-way switch - regular, Decora, etc. The most important thing is running 14/3 (or 12/3 on a 20A circuit) between the switches instead of 14/2, but you did that.
There are a few different ways that these switches can be wired. This is specific to your particular configuration. Note that there is no standard for where the screws are on switches! On a standard switch, it doesn't matter which wire goes to which screw. on a 3-way switch, it does matter. The instructions should make it clear, but pretty universal is that there will be different colors - one screw (= common) silver and two screws brass (= travelers) (or vice versa). I will refer to them as common and travelers, as that is what matters, no location or color.

Original switch box:

Old Black = hot = common screw on replacement 3-way switch.

Old Red = switched hot = white wire coming from new switch box.

Old White = neutral = still passes through, as currently connected to other neutrals.

New Black = traveler = traveler screw on replacement 3-way switch.
New Red = traveler = traveler screw on replacement 3-way switch.



New switch box. Replace standard switch with 3-way switch as follows:

Black = traveler screw

Red = traveler screw

White = common screw




White vs. Neutral
There is one more complication. Neutral and white are normally the same thing. A switch loop is an exception. In recent versions of the NEC, most switch boxes need to have neutral in a box, for new work. With a regular switch loop (one switch), the solution is to use /3 cable instead of /2, so that you have Black = hot, Red = switched hot, White = neutral. But with a 3-way switch, you need 3 wires so that means going to /4 cable, which is not nearly as common. If this were an existing circuit then this would not be an issue. But the 3-way 2nd-switch is brand new here, so grandfathering does not solve the problem.
I think you are OK here because one of the switch boxes in the circuit has neutral. But I am not 100% certain. If it turns out that you need a neutral in both boxes then you will either need /4 cable or use smart switches instead.
Assuming that you are allowed to use white as a common switched-hot wire, then you need to mark both ends of the wire (e.g., colored, but not green or white, electrical tape) to make it clear that this white wire is not neutral.