Do you really need a satellite dish to communicate with satellites?
No, if that were true, your phone's GPS would be rather unwieldy, wouldn't it? And satellite phones would be pretty impractical. (Satphones predate modern smartphones solidly.)
All that a satellite dish is an antenna with a very high directive gain. So, if your link budget says you don't need that, you can use a smaller antenna.
No general answer can be given – the amount of directivity you need is defined by the link budget, and the type of antenna you need to achieve that directivity is defined by the wavelength of your transmission.
Considering your previous question, I think I'd recommend looking into what a link budget is. To give you an idea, we can do a quick rough calculation based on purely fictive numbers.
- \$\text{SNR}_\text{min}\$: APRS required SNR at receiver: 10 dB
- \$b\$: APRS signal bandwidth: 1 kHz
- \$\text{NF}\$: Receiver Noise Figure: 4 dB
- \$N_0\$: Thermal Noise Density: -174 dBm/Hz (at 20 °C)
Required receive signal power at receiver: (-184 dBm/Hz + 30 dB Hz) + 10 dB = -144 dBm
- Operational Carrier Frequency: 10 GHz
- Transmitter – Satellite Relay free space distance 500 km
- Satellite Relay – Receiver free space distance 500 km
Free space path loss per direction:
\begin{align}
\operatorname{FSPL}(\text{dB})
&= 10\log_{10}\left(\left(\frac{4\pi d f}{c}\right)^2\right) \\
&= 20\log_{10}\left(\frac{4\pi d f}{c}\right) \\
&= 20\log_{10}(d) + 20\log_{10}(f) + 20\log_{10}\left(\frac{4\pi}{c}\right) \\
&= 20\log_{10}(500\text{ km}) + 20\log_{10}(10\text{ GHz}) - 147.55\\
&= 20\cdot 5.7 + 20\cdot 10 - 147.55\\
&= 114 + 200 - 147.55\\
&= 366.45
\end{align}
- Satellite Down- and Uplink antenna gains: 13 dB
- Relay amplification on the satellite: 60 dB
- Output power of your transmitter: 60 dBm (that's 1 freaking kilowatt at 10 GHz; that's nation-state level stuff)
Let the ground station antenna gains be undetermined, called \$G\$ at this point:
\begin{align}
-144\text{ dBm} &\overset!\le 60 \text{ dBm} + G_\text{TX} - \text{FSPL}_\text{uplink} + G_\text{Sat, uplink} + A_{sat} + G_\text{Sat, downlink} - \text{FSPL}_\text{downlink} + G_\text{RX} \\
&=\left(60 + 2\cdot 13 - 2\cdot 366.45 + 60\right)\text{ dBm} + G_\text{TX} + G_\text{RX}\\
&=\left(60 + 26 - 732.9 + 60\right)\text{ dBm} + G_\text{TX} + G_\text{RX} \\
&\approx -600 dBm + G_\text{TX} + G_\text{RX} \\
G_\text{TX} + G_\text{RX} &= 356 \text{ dB}
\end{align}